3. Water and Energy Budget Equations

The climate system is comprised of various subsystems such as the atmosphere, ocean and the land surface (including the biosphere, the geosphere and cryosphere). The variation and evolution of the climatic state of a region (as determined by climate variables such as temperatures, precipitation, humidity, soil moisture, etc.) is governed by the exchange of mass and energy through the boundaries of the region and between its climate subsystems, and the conversion of the mass and energy in their various forms within the system. For terrestrial regions such as the MRB, the climate subsystems of interest are naturally the atmosphere and the underlying land surface. Depending on the degree of details that is desired in the analysis, there are many forms of the water and energy conservation equations that could be used in the budget study. In this preliminary study, we will limit the analysis to the 2-dimensional (vertically-integrated) horizontal variations of key water and energy processes and adopt the set of budget equations (3.1)-(3.4) from Roads et al. (2002, 2003):

3.1 Atmospheric Water

Q t E P MC RESQ

Figure 3.1 Atmospheric Water Equation
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3.2 Surface Water

W t P E N RESW

Figure 3.2 Surface Water Equation
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3.3 Atmospheric Energy

Cp T t QR LP SH HC REST

Figure 3.3 Atmospheric Temperature Equation
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3.4 Surface Energy

Cv Ts t QRS LE SH RESG

Figure 3.4 Surface Temperature Equation
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3.5 Illustrations

Q time MC P E MC W time N

Figure 3.5 Water Cycle
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Schematic illustration of the water balance in the atmospheric and at the surface as represented in Eqs. 3.1 and 3.2

Cp T time QR HC HC Cv Ts time QRS LP LE SH

Figure 3.6 Energy Cycle
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Schematic illustration of the energy balance in the atmospheric and at the surface as represented in Eqs. 3.3 and 3.4

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3.6 Variable Descriptions

Q = Atmospheric Precipitable Water, mm
W = Surface Water (M + S), mm
M = Soil Moisture, mm
S = Snow, mm
T = Atmospheric Temperature, K
Ts = Surface Skin Temperature, K
T2 = Surface Air Temperature (at 2m), K
E = Evaporation, mm/day
P = Precipitation, mm/day
MC = Moisture Convergence, mm/day
N = Runoff, mm/day
LP = Latent Heat of Condensation, W/m^2, K/day
SH = Sensible Heat (which is positive upward), W/m^2, K/day
HC = Dry Static Energy Convergence, W/m^2, K/day
LE = Latent Heat of Evaporation (which is positive upward), W/m^2, K/day
QR = Atmospheric Radiative Heating (which is negative), W/m^2, K/day
QRS = Surface Radiative Heating, W/m^2, K/day
BOA SWU = Upward Shortwave Radiation at the Bottom of Atmosphere, W/m^2, K/day
BOA SWD = Downward Shortwave Radiation at the Bottom of Atmosphere, W/m^2, K/day
BOA LWU = Upward Longwave Radiation at the Bottom of Atmosphere, W/m^2, K/day
BOA LWD = Downward Longwave Radiation at the Bottom of Atmosphere, W/m^2, K/day
TOA SWU = Upward Shortwave Radiation at the Top of Atmosphere, W/m^2, K/day
TOA SWD = Downward Shortwave Radiation at the Top of Atmosphere, W/m^2, K/day
TOA LWU = Upward Longwave Radiation at the Top of Atmosphere, W/m^2, K/day
RESQ = Atmospheric Water Residual Forcing, mm/day
RESW = Surface Water Residual Forcing, mm/day
REST = Atmospheric Dry Static Energy Residual Forcing, W/m^2, K/day
RESG = Surface Temperature Residual Forcing, W/m^2, K/day

{ } denotes vertical integrals over either the atmospheric column or soil column

Detailed derivations and discussions of the budget equations, as well as the computational details of the budget terms can be found in Roads et al. (2002, 2003; see also GCIP WEBS) and will not be repeated here. The main assumptions employed include the neglect of kinetic energy in the conservation of atmospheric energy and the neglect of latent heat of fusion during the formation of ice-phase precipitation in the atmosphere and during the snowmelt at the surface. Further discussions on the latter effects on the basin energy budgets can be found in the discssuion of surface snowcover to be presented later. Vertical integrations of atmospheric quantities are evualted by using the method outlined in Trenberth (1991). Similar to Roads et al. (2002, 2003), the vertically-integrated water budget terms (kg/m2s) are multiplied by 8.64 x 10^4 s/day to provide values in kg/(m2 day) or mm/day; and all energy flux terms (W/m2) are multiplied by (8.64 X 10^4 s/day)/(Cp Ps/g), where Ps = monthly surface pressure, to provide normalized values in units of K/day. The surface energy terms are also multiplied by a constant atmospheric normalization (i.e., CvH=Cp(Ps/g), where H = depth of soil layer) in order to provide values in K/day so that they can easily be compared to the atmospheric counterparts. It is convenient to combine the residual forcings with the tendency terms in the budget results (e.g. RESW = RESW-dW/dt). This should have little effects on the values of the mean annual residues since, over a long period, the change in water and energy storages in the atmosphere and surface can be assumed to be negligible. However, it should be noted that the storage terms, at least the surface storage terms, can exhibit strong seasonal variations over this vast northern terrestial region. As this set of budget equations and normalizations are also used in the GCIP WEBS (Roads et al., 2003), their adoption for use in this study will facilitate the intercomparisons of budget results from different CSE regions in GHP-wide or other global WEBS efforts.

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