## The heat budget equation

The heat budget of the northern polar cap can be approximated as:

Equation (3.3) states that the time change in the storage of moist static energy (defined in the introduction) in the Arctic atmosphere AE/At is represented by the sum of the

f wall \25mb

Figure 3.5 Schematic of the energy balance for the north polar cap (from Nakamura and Oort, 1988, by permission of AGU).

f wall \25mb

net radiation at the top of the atmosphere (Frad), the net poleward energy flux across a hypothetical wall at 70° N extending from the surface to the top of the atmosphere (Fwall) and the net heat flux at the Earth's surface (Fsfc). If the sum of the three terms on the right is positive, the atmosphere gains energy. If their sum is negative, the atmosphere loses energy. All units are in W m-2. Figure 3.5 provides a schematic of the problem. The basic question we are asking from Equation (3.3) is: what are the relative contributions of the net fluxes of energy into the sides, top and bottom of the polar cap and how do these vary seasonally to cause changes in the energy content of the atmosphere? By answering this question, we can learn a great deal about how the Arctic functions. Analogous approaches to those described here can be used to compile gridded fields of energy budgets across the globe, as done by Trenberth et al. (2001). Consider first the moist static energy content, E, of the atmosphere at a given time. This is represented as:

The terms on the right, respectively, represent the energy content of the atmosphere in the form of internal energy (sensible heat content), latent heat and potential energy. Here, CP is the specific heat of the atmosphere at a constant pressure (1005.7 J K-1Kg-1), T is temperature in kelvin, L is the latent heat of evaporation (2.5 x 106 J kg-1), q is specific humidity (g kg-1), g is gravitational acceleration (9.8 m s-2), and z is geopotential height (gpm). Hence, moist static energy increases with a rise in temperature, a rise in moisture content and a rise in geopotential (gz = \$). The geopotential represents the work required to raise a unit mass to a height z from sea level: the greater the geopotential, the greater the potential energy.

Each of the three terms in parentheses has units of joules per kilogram (J kg-1), i.e., energy per unit mass. The term gz has the more obvious units of m2 s-2. However, energy (J) has units of kg m2 s-2. Rearranging for m2 and substituting into m2 s-2 yields the desired units of J kg-1. The total energy content of the polar cap, E, has units of joules, and is obtained by integrating over the volume (dV). Note that in the vertical dimension, extending from the surface to the top of the atmosphere, there must be a mass weighting, accomplished through integration by dp/g (units of kg m-2), where p is pressure.

Nakamura and Oort (1988) obtained mean monthly values of E from a 10-year data set compiled by the Geophysical Fluid Dynamics Laboratory (GFDL), based mainly on rawinsonde data and extending to the 25 hPa level (hence containing most of the atmospheric mass). Monthly values of AE/At were found from the difference in E between neighboring months. For example, as E is greater in May than in April, for May AE/At is positive. The resulting values in units of joules per unit time, divided by the area of the polar cap, yield the final desired units of W m-2.

Frad, the net radiation at the top of the atmosphere, can be broken down as:

Fsw is the area-averaged downward TOA shortwave (solar) flux, which varies seasonally with respect to solar declination. A is the area-averaged albedo for the polar cap of the Earth-atmosphere system (i.e., the planetary albedo), which as discussed depends on the surface albedo, cloud cover, and clear-sky scattering and absorption. Flw is the longwave radiation emitted to space, due to emission from both the atmosphere and the surface. Nakamura and Oort (1988) based Frad on reflected solar and outgoing longwave fluxes measured by Earth-orbiting satellites between the years 1966 and 1977.

Fwall represents the net horizontal transport of moist static energy into the polar cap. It can be broken down as follows:

íwaii = JJCp[ vT ]dx dp/g + JJL[ vq ]dx dp/g + JJg[ vz ]dx dp/g (3.6)

where overbars denote time means and the brackets [ ] denote zonal means. The three terms on the right are the fluxes of: (1) sensible heat, (2) latent heat, and (3) potential energy. If their sum is positive (more energy comes into the sides of the column than leaves), the total flux is positive. The term [vT] is the zonal mean of the time-averaged meridional temperature flux, where v is the meridional (north-south) component of the wind. Multiplication by cp, yields the zonal-mean, time-averaged meridional sensible heat flux. This zonal-mean term is evaluated at pressure levels from the surface to the top of the atmosphere. The zonal-mean values at each level are then integrated around 70° N (dx, with x being a unit distance). The integrals at each level are then integrated vertically (dp/g). In a similar vein, the meridional latent heat flux is calculated from horizontal and vertical integration of the time-mean, zonal-mean transport of latent heat. Finally, the flux of potential energy is obtained from the zonal mean of the time-mean product of the meridional wind and geopotential height. Nakamura and Oort (1988) assessed Fwall and its components with the 10-year GFDL data set.

The final term in Equation (3.3) is the net surface flux. This is the net heat transfer between the atmospheric column (bounded by 70° N) and a column (also bounded by 70° N) extending from the surface downward through the land and ocean. We can term the latter the ocean-ice-land column. Note that 72% of the surface area of the polar cap is underlain by ocean. Considered in isolation, a net transfer of heat from the ocean-ice-land column into the atmospheric column means that the atmospheric column gains heat while the ocean-ice-land column loses an equivalent amount. Conversely, if there is a net transfer of heat from the atmospheric column into the ocean-ice-land column, the atmospheric column loses heat while the ocean-ice-land column gains an equivalent amount. The term hence represents the net reservoir (or volume) exchange. The net surface heat flux should not be confused with the energy budget at the surface interface (Chapter 5), which must always sum to zero. In the absence of reliable surface data, Nakamura and Oort (1988) obtained the net surface heat flux as a residual of the other terms in Equation (3.3).

Before continuing some clarifications are warranted regarding moist static energy (Equation (3.4)) and its changes (Equation (3.3)). Moist static energy is not the total energy content of the atmosphere. It does not include the small sensible heat content of water mass in liquid or solid forms in the atmosphere (in clouds or falling from the atmosphere as rain or snow) nor does it include kinetic energy (the energy of motion). However, as time changes in these terms are generally quite small, we can ignore them. Furthermore, moist static energy, as defined, considers the liquid phase of water to be the zero latent heat state. For "bookkeeping", this means that, considered in isolation, a release of heat to the atmospheric column associated with fusion (3.34 x105 Jkg-1) would be counted as a negative latent heat flux to the surface (as snowfall). This effect would in turn count as a net surface flux from the ocean-ice-land column into the atmospheric column.

To help further clarify the net surface flux, it is useful to develop an expression for the energy budget of the ocean-ice-land column similar to Equation 3.3:

The term on the left is the rate of storage of heat in the ocean-ice-land column. The first term on the right, Fo, is the net poleward oceanic heat transport. A net poleward (equatorward) flux of oceanic heat across 70° N acts to increase (decrease) the heat content of the ocean-ice-land column. Fi is also a horizontal transfer, and represents the net poleward flux of latent heat in the form of snow and ice. As introduced in Chapter 2 and expanded upon in Chapter 7, there is net outflow of sea ice (and overlying snow cover) from the polar cap into the North Atlantic. A given mass of ice has a lower energy content than the same mass of water at the same temperature. Assuming that the exported snow and ice is replaced by an equivalent mass of water at the same temperature, this outflow represents an effective increase in the heat content of the ocean-ice-land column. Equations (3.7) and (3.3) are linked through the same surface heat flux term. Building on previous discussion, it should be evident that if Fsfc in Equation (3.7) is positive (a flux from the ocean-ice-land column into the atmospheric column), this contributes to a loss of heat by the ocean-ice-land column, which must be seen in Equation (3.3) as a gain of heat by the atmospheric column. If Fsfc in

Equation (3.7) is negative, it contributes to a loss of heat in the atmospheric column, balanced by a gain of heat in the ocean-ice-land column. Equation (3.7) assumes that net horizontal heat transfers into the column via non-oceanic process can be ignored. It also assumes that the column extends to a sufficient depth so that we can ignore heat transfers into its bottom.

Over the long-term annual mean, the change in energy storage in Equation (3.7) (AE * /AT) is zero. Hence, the annual net surface heat flux must equal the heat flow across the boundaries of the ocean-ice-land column represented by Fo and Fi. Naka-mura and Oort (1988) computed (as a residual of the other terms in Equation (3.3)) an annual mean surface flux of 2.4 W m-2. An estimate of Fi was obtained based on an estimated annual flow of ice out of the Arctic of 3 x 1015 kg yr-1. As averaged over the area north of 70° N, this is equivalent to an annual heating of only 2.1 W m-2. Reiterating previous discussion, this assumes that the exported ice is replaced by the same amount of water at the same temperature. A recent estimate of the annual ice outflow through Fram Strait by Vinje (2001) of about 2.6 x 1015 kgyr-1 corresponds to about 1.8 W m-2 (assuming that all the ice exits the polar cap without melting). The flux of ice through the channels of the Canadian Arctic Archipelago would represent a small additional contribution. From their assumed annual mean Fsfc of 2.4 W m-2, and their annual mean Fi of 2.1 W m-2, Nakamura and Oort (1988) estimated a value for Fo of only 0.3 W m-2. The small annual mean ocean heat transport in polar latitudes from this residual estimate is consistent with Figure 3.3. Aagaard and Greisman (1975) estimate that the ocean heat transports are about 0.002 and 0.14 PW respectively through the Bering Strait and the North Atlantic (at about 65° N). These compare to maximum values in Figure 3.3 of about 2 PW around 17° N.

Simply having a small annual mean of Fsfc does not imply that it is negligible for a given month. As will be shown, the net surface flux for a given month can differ from the annual mean value by over a factor of 40. However, Nakamura and Oort (1988) made the assumption that while Fo and Fi are small in the annual mean sense, they are also small and relatively steady through the year. To a reasonable approximation, these two terms can hence be dropped from Equation (3.7) for analysis of the annual cycle. There is actually about a factor of three seasonal range for the mean Fram Strait ice transport (see Chapter 7). While the assumed steady nature of Fi is hence not entirely valid, we are still led to conclude that while sea ice transport is extremely important for the freshwater budget of the Arctic Ocean (it represents the primary sink), it has a relatively small influence on the heat budget of the ocean-ice-land column.

With the assumption that net horizontal heat transfers into the ocean-ice-land column can be ignored, the net surface heat flux can be expressed as:

Sm represent the rate of storage (S) of latent heat in the form of snow and ice. The formation of sea ice counts as a gain of heat by the atmospheric column and a corresponding loss from the ocean-ice-land column. From our convention, snowfall acts in the same sense. In both cases, Shu would be negative. As examined in Chapters 5 and 7, most sea ice growth occurs at the underside of existing ice (at the ice-ocean interface). The heat released by ice formation is then conducted upwards to the surface and out of the column. If SM is positive, this implies the surface melt of sea ice and snow, counting as a gain of heat by the ocean-ice-land column and a loss from the atmospheric column. So is the rate of storage of sensible heat in the ocean. If the term is positive, the ocean is gaining sensible heat (its temperature rises), which represents a heat loss from the atmosphere. If the term is negative, the ocean is losing heat, and the atmosphere is gaining heat. Sl is the rate of storage of sensible heat of the land. A gain of sensible heat over land means an addition of heat to the subsurface. As this is heat that could otherwise be used to increase the temperature of the atmosphere, positive Sl has a negative effect on the atmospheric heat content. The opposing effect holds for when Sl is negative. Si is the rate of storage of sensible heat in snow, which is interpreted similarly.

Simplification is again possible as Sl and Si appear to be relatively small (Nakamura and Oort, 1988). Hence, the net surface heat flux is primarily contained in the growth and melt of sea ice, snow cover and the change in sensible heat storage of the ocean. Nakamura and Oort (1988) were able to estimate monthly values of So based on the Levitus (1984) analysis of the heat content of the global oceans. With this estimate of So and the residual term Fsfc, they were then able to estimate Sm as another residual.

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