Heat Budget

Overview

The objective of this section is to examine the model-calculated heat budget terms on Lake Folsom and verify the conservation of energy equation. A test simulation was performed using the full American River network (Figure 1), with water temperature as the only active constituent.

Figure 1- Full lower American watershed, containing Lake Folsom, Lake Natoma, and the Lower American River

Model Set-up

The simulation period was calendar year 2014. Lake Folsom was given an isothermal initial condition of 10°C. Inflows, reservoir operations, boundary conditions, and meteorological data were based on observed data for the watershed. Figure 2 shows the meteorological data records as an example.

Figure 2 - Meteorological data

To verify the heat budget calculations, the heat content of the reservoir was calculated as:

$\begin{array}{l}\displaystyle H_{Folsom}= \rho C_p \sum_{k} V_k T_k\end{array}$

Where:

$\begin{array}{l}H_{Folsom}\end{array}$ is the heat content of Lake Folsom (J),
$\begin{array}{l}V_k\end{array}$ is the computational layer volume (m³),
$\begin{array}{l}T_k\end{array}$ is the temperature of the layer (°C),
$\begin{array}{l}\rho\end{array}$ is the water density (assumed = 1000 kg/m³), and
$\begin{array}{l}C_p\end{array}$ is the specific heat of water (assumed = 4186 J/kg/°C).

The change in heat content from time step to time step ( $\begin{array}{l}ΔH_{Folsom}\end{array}$ ) was compared to the net boundary heat fluxes for the reservoir. These include inflow, outflow, and meteorological fluxes (shortwave radiation, longwave radiation, evaporative, and sensible). Heat fluxes associated with inflows and outflows ( $\begin{array}{l}F_Q\end{array}$ ) were calculated as:

$\begin{array}{l}\displaystyle F_Q=QρC_P T_Q\end{array}$

Where:

$\begin{array}{l}Q\end{array}$ is the inflow or outflow magnitude (m³/s), and

$\begin{array}{l}T_Q\end{array}$ is the temperature of the flow (°C).

Net meteorological fluxes ( $\begin{array}{l}F_{met}\end{array}$ ) were calculated by using the heat fluxes output from the Water Temperature Simulation Module (WTSM) (in W/m²) and multiplying by the layer top area ( $\begin{array}{l}A_k\end{array}$ ). For the surface layer, the surface area of the reservoir was used; for all deeper layers, the area was the top area of each computational layer.

$\begin{array}{l}\displaystyle F_{met} = \sum_{k} \big( q_{sw,k}+q_{atm,k}-q_{b,k}+q_{h,k}-q_{l,k}+q_{sed,k} \big) A_k\end{array}$

Longwave ( $\begin{array}{l}q_{sw} , q_{atm}\end{array}$ ), sensible ( $\begin{array}{l}q_h\end{array}$ ), and latent ( $\begin{array}{l}q_l\end{array}$ ) heat fluxes were only non-zero in the surface layer.

A plot of the individual terms in the heat budget is shown in Figure 3 for a week in early July.

Figure 3 - Individual terms in the heat budget - July

To check closure of the heat budget, the residual between the change in heat in Lake Folsom and the sum of the boundary heat fluxes was calculated as:

$\begin{array}{l}\displaystyle Residual=(∆H_{Folsom}-(F_{Qin}-F_{Qout}+F_{met} ))∆t\end{array}$

where $\begin{array}{l}∆t\end{array}$ is the computational time step (sec).

A plot of the hourly heat fluxes is shown in Figure 4 and Figure 5. Figure 4 shows a few days in July to see the diurnal variations. Figure 5 shows the full period. Below each heat flux plot is the residual time series. Residuals were approximately 1 x 10^-8, indicating a successful closure of the heat budget.

Figure 4 - Hourly heat fluxes - July

Figure 5 - Hourly heat fluxes - Full period