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Temperature Index Method
Overview
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 USACE in the 1950s (USACE 1956).
Basic Concepts
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):
Melt = DryMeltRate * (Air Temperature - Base Temperature) |
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):
Melt = (WetMeltRate + 0.168 * PrecipitationIntensity) * (Air Temperature - Base Temperature) |
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
\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) |
where Qt = 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); Tat = the air temperature; Ts = a representative temperature of the snow pack; and cr = 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, Ts 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, defined and estimated as (Anderson 1973, Corps of Engineers 1987, p 18)
ATICC_2=ATICC_1+TIPM \cdot (T_a-ATICC_1) |
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
\rho_{s} C_{p} d \frac{\partial T_{A T I}}{\partial t}=h^{*}\left(T_{a}-T_{A T I}\right) |
where d = the “depth” of the snow pack associated with the depth of the index temperature; TATI = 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 Ta = the air temperature. Note that we are assuming that a region of the snow pack temperature has a uniform temperature TATI. This assumption is a bit dubious BUT it makes TATI entirely analogous to ATICC. Note also that h* can be defined as
h^{*}=\frac{1}{\frac{1}{h}+\frac{l_{s}}{k_{s}}} |
where h = the heat transfer coefficient from the snow surface to the atmosphere; ks = the snow thermal conductivity; and ls = 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 ls. If it is assumed that the snowpack temperature is To at time t = 0, the solution for equation is
T_{A T I}=T_{a}+\left(T_{o}-T_{a}\right) e^{-\frac{h^{*}}{\rho_{s} C_{p} d} t} |
where To = the initial snow pack temperature; and t = time from start. Setting TATI ATICC2 and To ATICC1, equation (6) can be restated as
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) |
By comparing equation and equation , the expression for TIPM results:
TIPM= \left(1-e^{-\frac{h^{*}}{\rho_{s} C_{p} d} t}\right) |
We can see that TIPM is a function of the material properties of the snow pack, ρs, cp, 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
TIPM_2=1-(1-TIPM_1)^{\frac{t_2}{t_1}} |
where t2 = the time step corresponding to TIPM2. Equation was also presented without explanation in Anderson (2006). Equation can now be restated as
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) |
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
\frac{\partial c c}{\partial t}=c_{r}\left(T_{a}-T_{A T I}\right) |
where cc = cold content (inches day-1); and cr = cold rate (in. day-1 °F-1). Equation can be integrated by again setting TATI ATICC2; noting the solution for ATICC2 as given in equation to arrive at
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) |
Where log = the natural logarithm; and, as before, TIPM1 is the value of TIPM calibrated 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 cr.).
Required Parameters
The PX Temperature is used to differentiate between precipitation falling as rain or snow. In particular, precipitation that falls at an air temperature above the PX Temperature will occur purely as rain while precipitation that falls at an air temperature below the PX Temperature will occur purely as snow. Decreasing the PX Temperature will cause more precipitation to fall purely as rain while increasing the PX Temperature will cause less precipitation to fall purely as rain.
The Base Temperature is the temperature above which snow begins to melt. This parameter typically has a value around the freezing temperature, but can vary by a few degrees. Decreasing the Base Temperature will cause snow melt to occur at colder temperatures while increasing the Base Temperature will require higher temperatures to cause snow melt.
The ATI Coefficient is used to weight the previous time step's ATI in the computation of the current time step's ATI.
The Wet Melt Rate represents the rate at which the snowpack melts when it is raining on the snowpack and air temperatures are in excess of the Base Temperature. Increasing the Wet Melt Rate will increase the rate of snow melt while decreasing the Wet Melt Rate will decrease the rate of snow melt.
The Rain Rate Limit is used to differentiate between wet and dry melt rates. If the daily precipitation rate exceeds this value, the wet melt rate will be used.
The Dry Melt Rate represents the rate at which the snowpack melts when it is not raining on the snowpack and air temperatures are in excess of the Base Temperature. Increasing the Dry Melt Rate will increase the rate of snow melt while decreasing the Dry Melt Rate will decrease the rate of snow melt.
The Cold Limit is used to reset the cold content. When the daily precipitation rate exceeds this value, the antecedent cold content index is set to the temperature of the precipitation. If the temperature is above the base temperature, the cold content index is set to the base temperature. If the temperature is below the base temperature, the cold content index is set to the actual temperature. If the precipitation rate is less than the cold limit, the cold content index is computed as an antecedent index.
The ATI-Coldrate Function is used to calculate a cold content from the current cold content index.
The Coldrate Coefficient is used to update the antecedent cold content index from one time interval to the next. This is a separate index from the one used to update the meltrate index.
The Water Capacity defines the liquid water content above which water leaves the snowpack. Increasing the Water Capacity will delay the time at which water leaves the snowpack and vice versa.
The Groundmelt represents the rate at which the snowpack melts due to energy from the ground. Increasing the Groundmelt will increase the rate of snow melt while decreasing the Groundmelt will decrease the rate of snow melt.
The following table presents units, a summary description, allowable values within HEC-HMS, and a recommended range for each of the aforementioned parameters.
Parameter Name | Units | Description | Allowable Range (min - max) | Recommended Range (min - max) |
---|---|---|---|---|
PX Temperature | deg F deg C | Temperature above which precipitation falls as rain | 20.3 - 45.23 deg F -6.5 - 7.5 deg C | 32 - 37 deg F 0 - 2.78 deg C |
Base Temperature | deg F deg C | Temperature above which energy is added to the snowpack | 20.3 - 45.23 deg F -6.5 - 7.5 deg C | 32 - 35 deg F 0 - 1.67 deg C |
ATI Coefficient | - | Used to weight the previous time step's ATI in the computation of the current time step's ATI | 0 - 1.0 | 0.98 |
Wet Meltrate | in/deg F-day mm/deg C-day | Represents the rate at which the snowpack melts when it is raining on the snowpack and air temperatures are in excess of the Base Temperature | 0.0 - 2.19 in/deg F-day 0 - 100 mm/deg C-day | 0.05 - 0.1 in/deg F-day 2.29 - 4.57 mm/deg C-day |
Rain Rate Limit | in/day mm/day | Precipitation rate threshold that discriminates between dry melt and wet melt | 0 - 236.2 in/day 0 - 6000 mm/day | 0.1 in/day 2.54 mm/day |
Dry Melt Rate / ATI-Meltrate Function | in/deg F-day mm/deg C-day | Represents the rate at which the snowpack melts when it is not raining on the snowpack and air temperatures are in excess of the Base Temperature | 0.0 - 0.22 in/deg F-day 0 - 10 mm/deg C-day | 0.05 - 0.1 in/deg F-day 2.29 - 4.57 mm/deg C-day |
Cold Limit | in/day mm/day | Precipitation rate threshold for resetting cold content | 0 - 236.2 in/day 0 - 6000 mm/day | 0.0 - 0.5 in/day 0.0 - 12.7 mm/day |
ATI-Coldrate Function | in/deg F-day mm/deg C-day | Used to calculate a cold content from the current cold content index | 0.0 - 0.22 in/deg F-day 0 - 10 mm/deg C-day | 0.01 - 0.03 in/deg F-day 0.46 - 1.38 mm/deg C-day |
Coldrate Coefficient | - | Used to update the antecedent cold content index from one time interval to the next | 0 - 0.99999 | 0.35 - 0.5 |
Water Capacity | % | Liquid water content above which water leaves the snowpack | 0 - 100.0 | 3 - 5% |
Groundmelt | in/day mm/day | Represents the rate at which the snowpack melts due to heat from the ground | 0 - 0.39 in/day 0 - 10 mm/day | 0 in/day 0 mm/day |
A Note on Parameter Estimation
Regardless of the source of information used to estimate initial parameter values, including the table presented above, all Hybrid snow parameters must be calibrated and validated.
Required Boundary Conditions
In addition to parameters, the Hybrid snow method requires the following meteorologic boundary conditions:
Tutorials describing example applications of this snowmelt method, including parameter estimation and calibration, can be found here: Calibrating Gridded Snowmelt: Upper Truckee River, California and Calibrating Point Snowmelt: Swamp Angel Study Plot, Colorado.
A tutorial illustrating how to use the Uncertainty Analysis to evaluate Hybrid Snow parameter sensitivity can be found here: Evaluating Temperature Index Snowmelt Parameter Sensitivity (Continuous Simulation) and Evaluating Temperature Index Snowmelt Parameter Sensitivity (Event Simulation).
Descriptions of the user interface features pertaining to this method can be found here: Gridded Temperature Index Snow section of the User's Manual.