Temperature Index Method

The HEC-HMS Temperature Index Snow Model is described in this section. This temperature index approach is derived, almost directly, from the Temperature Index Snow Model of SSARR model (USACE 1987). The SSARR Temperature Index Snow Model was based, in large part, on the original snow hydrology studies conducted by the Corps of Engineers in the 1950s (USACE 1956).

Overview

Mass Balance

A temperature index snow model simplifies the heat transfer calculations into the snowpack by estimating the heat transfer into or out of the snowpack as a function of the difference between the surface temperature and the air temperature.

During dry melt conditions, the program uses the following equation to compute snowmelt (assuming no cold content in the snow pack):

$\begin{array}{l}\displaystyle Melt = DryMeltRate * (Air Temperature - Base Temperature)\end{array}$

where DryMeltRate is in inches/(Degree Fahrenheit-Day) or mm/(Degree Celsius-Day).

During rain on snow conditions, the program uses the following equation to compute snowmelt (assuming no cold content in the snow pack):

$\begin{array}{l}\displaystyle Melt = (WetMeltRate + 0.168 * PrecipitationIntensity) * (Air Temperature - Base Temperature)\end{array}$

where WetMeltRate is in inches/(Degree Fahrenheit-Day) or mm/(Degree Celsius-Day) and the constant term, 0.168, has units of hour/(Degree Fahrenheit-Day) or hour/(Degree Celsius-Day). The rain on snow equation is based on equation 5-18 in Engineering Manual 1110-2-1406.

The snowmelt capability in HEC-HMS estimates the following snowpack properties at each time step: The Snow Water Equivalent (SWE) accumulated in the snowpack; the snowpack temperature (actually, the snowpack cold content but this is equivalent to the snowpack temperature); snowmelt (when appropriate); the liquid water content of the snowpack; and finally, the runoff at the base of the snowpack.

Energy Balance

Cold Content

The rate of change of cold content with time can be approximated starting with the definition from equation as

$\begin{array}{l}\displaystyle \frac{d C_{c}}{d t}=\frac{Q_{t}}{\rho_{w} \lambda_{w}} \approx \frac{h\left(T_{a_{t}}-T_{s}\right)}{\rho_{w} \lambda_{w}}=c_{r}\left(T_{a_{t}}-T_{s}\right)\end{array}$

where Q_t = the rate of heat transfer per unit area (energy per unit area per time); h= a heat transfer coefficient (energy per unit area per time per degree air temperature); T_at = the air temperature; T_s = a representative temperature of the snow pack; and c_r = the “cold rate” that will be discussed below. Note that, following the example of Anderson (1973) and others, the engineering approximation of heat transfer has been used. There is a question of what the representative temperature of the snowpack should represent. To be entirely consistent with the concept of engineering heat transfer coefficient, T_s should equal the surface temperature of the snowpack. However, this is not very satisfactory because the surface temperature of the snow pack is not known a priori. This is because the heat transfer from the snow pack is controlled both by the heat transfer from the surface to the atmosphere and by the heat conduction through the snowpack itself; with the slower of the two processes controlling the rate. To overcome this problem, the representative temperature of the snow pack will be considered to represent some interior temperature of the snowpack. If the snowpack is shallow, the temperature will be representative of the entire snowpack; if the snowpack is deep, the temperature will be representative of the upper layer. This representative temperature, termed the “Antecedent Temperature Index for Cold Content” (ATICC) will be estimated using quasi-engineering approach to heat transfer in a somewhat similar manner as the cold content, as described below.

Index Temperature

The cold content is found by first estimating an “Antecedent Temperature Index for Cold Content” (ATICC) “near” the snow surface, ATICC, defined and estimated as (Anderson 1973, Corps of Engineers 1987, p 18)

$\begin{array}{l}\displaystyle ATICC_2=ATICC_1+TIPM \cdot (T_a-ATICC_1)\end{array}$

where ATICC₂ = the index temperature at the current time step; ATICC₁ = the index temperature at the previous time step; and TIPM is a non-dimensional parameter. The problem is that limited documentation exists to describe how the parameter TIPM is related to the time step, snow material properties, or heat transfer conditions.

In this section a consistent approach for estimating cold content is developed that is based on the approach of estimating changes in cold content based on the temperature difference between the air temperature and ATICC. First, an approach for estimating ATICC is developed. To do this, we turn to a simple heat budget type analysis of the snow pack in order to gain some insight. A straightforward heat budget of the snow pack can be written

$\begin{array}{l}\displaystyle \rho_{s} C_{p} d \frac{\partial T_{A T I}}{\partial t}=h^{*}\left(T_{a}-T_{A T I}\right)\end{array}$

where d = the “depth” of the snow pack associated with the depth of the index temperature; T_ATI = the snow temperature measured by the antecedent temperature index, h* = the “effective” heat transfer coefficient from the snow surface to the atmosphere (wm^-2°C^-1); and T_a = the air temperature. Note that we are assuming that a region of the snow pack temperature has a uniform temperature T_ATI. This assumption is a bit dubious BUT it makes T_ATI entirely analogous to ATICC. Note also that h* can be defined as

$\begin{array}{l}\displaystyle h^{*}=\frac{1}{\frac{1}{h}+\frac{l_{s}}{k_{s}}}\end{array}$

where h = the heat transfer coefficient from the snow surface to the atmosphere; k_s = the snow thermal conductivity; and l_s = the effective snow depth through which thermal conduction occurs. h* will be dominated by whichever process is slower: heat transfer from the snow to the atmosphere or thermal conduction through the snow depth l_s. If it is assumed that the snowpack temperature is T_o at time t = 0, the solution for equation is

$\begin{array}{l}\displaystyle T_{A T I}=T_{a}+\left(T_{o}-T_{a}\right) e^{-\frac{h^{*}}{\rho_{s} C_{p} d} t}\end{array}$

where T_o = the initial snow pack temperature; and t = time from start. Setting T_ATI ATICC₂ and T_o ATICC₁, equation (6) can be restated as

$\begin{array}{l}\displaystyle A T I C C_{2}=A T I C C_{1}+\left(1-e^{-\frac{h^{*}}{\rho_{s} C_{p} d} t}\right) \cdot\left(T_{a}-A T I C C_{1}\right)\end{array}$

By comparing equation and equation , the expression for TIPM results:

$\begin{array}{l}\displaystyle TIPM= \left(1-e^{-\frac{h^{*}}{\rho_{s} C_{p} d} t}\right)\end{array}$

We can see that TIPM is a function of the material properties of the snow pack, ρ_s, c_p, and d; the heat transfer regime as indicated by h*; and the time step, t. If we assume that the material properties and heat transfer regime of the snow pack are set by the value of TIPM corresponding to a given time step of one day (for example, TIPM₁ is the value of TIPM corresponding to a time step, t₁, of one day) then the value of TIPM at another time step with same material and heat transfer properties can be found as

$\begin{array}{l}\displaystyle TIPM_2=1-(1-TIPM_1)^{\frac{t_2}{t_1}}\end{array}$

where t₂ = the time step corresponding to TIPM₂. Equation was also presented without explanation in Anderson (2006). Equation can now be restated as

$\begin{array}{l}\displaystyle A T I C C_{2}=A T I C C_{1}+\left(1-\left(1-T I P M_{1}\right)^{\frac{t_{2}}{2}}\right) \cdot\left(T_{a}-A T I C C_{1}\right)\end{array}$

where TIPM₁ is the value of TIPM corresponding to a time step of one day; and t₂/t₁ is the ratio of the model time step (t₂) to one day (t₁). Equation employs the value of TIPM₁ calibrated from a time step of one day, and allows it to be used in model runs of 1 hour or even 1 minute and arrive at the same results for ATICC if the air temperature is the same.

If a simple differential equation for cold content is used

$\begin{array}{l}\displaystyle \frac{\partial c c}{\partial t}=c_{r}\left(T_{a}-T_{A T I}\right)\end{array}$

where cc = cold content (inches day^-1); and c_r = cold rate (in. day^-1°F^-1). Equation can be integrated by again setting T_ATI ATICC₂; noting the solution for ATICC₂as given in equation to arrive at

$\begin{array}{l}\displaystyle c c_{2}=c c_{1}-\frac{c_{r} t_{1}\left(1-\left(1-T I P M_{1}\right)^{\frac{t_{2}}{t_{1}}}\right)}{\log \left(1-T I P M_{1}\right)}\left(T_{a}-A T I C C_{1}\right)\end{array}$

Where log = the natural logarithm; and, as before, TIPM₁ is the value of TIPMcalibrated for a time step of one day; and t₂/t₁ is the ratio of the model time step (t₂) to one day (t₁). where t₂/t₁ is the ratio of the model time step (t₂) to one day (t₁) (or, more exactly t₁ should correspond to the units of c_r.).