8. Initial and Boundary Data

8.1. Initial Data

In this section, we describe how the time integration is started from data consistent with the spectral truncation. The land surface model requires its own initial data, as described by [Bon96]. The basic initial data for the model consist of values of u, v, T, q, \Pi, and \Phi_s on the Gaussian grid at time t=0. From these, U, V, T', and \Pi are computed on the grid using (101) , and (139) . The Fourier coefficients of these variables U^m, V^m, T'^m, \Pi^m, and \Phi^m_s are determined via an FFT subroutine (233) , and the spherical harmonic coefficients T'^m_n, \Pi^m_n, and \left(\Phi_s \right)^m_n are determined by Gaussian quadrature (234) . The relative vorticity \zeta and divergence \delta spherical harmonic coefficients are determined directly from the Fourier coefficients U^m and V^m using the relations,

\begin{aligned} \zeta & = \frac{1}{a(1 - \mu^2)} \frac{\partial V}{\partial \lambda} - \frac{1}{a} \frac{\partial U}{\partial \mu} , \\ \delta & = \frac{1}{a(1 - \mu^2)} \frac{\partial U}{\partial \lambda} + \frac{1}{a} \frac{\partial V}{\partial \mu} . \end{aligned}

The relative vorticity and divergence coefficients are obtained by Gaussian quadrature directly, using (237) for the \lambda –derivative terms and (240) for the \mu –derivatives.

Once the spectral coefficients of the prognostic variables are available, the grid–point values of \zeta, \delta, T', \Pi, and \Phi_s may be calculated from (263) , the gradient \nabla \Pi from (266) and (267) , and U and V from (271) and (272) . The absolute vorticity \eta is determined from the relative vorticity \zeta by adding the appropriate associated Legendre function for f (203) . This process gives grid–point fields for all variables, including the surface geopotential, that are consistent with the spectral truncation even if the original grid–point data were not. These grid–point values are then convectively adjusted (including the mass and negative moisture corrections).

The first time step of the model is forward semi–implicit rather than centered semi–implicit, so only variables at t=0 are needed. The model performs this forward step by setting the variables at time t = -\Delta t equal to those at t = 0 and by temporarily dividing 2 \Delta t by 2 for this time step only. This is done so that formally the code and the centered prognostic equations of chapter [chap3] also describe this first forward step and no additional code is needed for this special step. The model loops through as indicated sequentially in chapter [chap3]. The time step 2 \Delta t is set to its original value before beginning the second time step.

8.2. Boundary Data

In addition to the initial grid–point values described in the previous section, the model also requires lower boundary conditions. The required data are surface temperature (T_s) at each ocean point, the surface geopotential at each point, and a flag at each point to indicate whether the point is land, ocean, or sea ice. The land surface model requires its own boundary data, as described by [Bon96]. A surface temperature and three subsurface temperatures must also be provided at non-ocean points.

For the uncoupled configuration of the model, a seasonally varying sea–surface temperature, and sea–ice concentration dataset is used to prescribe the time evolution of these surface quantities. This dataset prescribes analyzed monthly mid-point mean values of SST and ice concentration for the period 1950 through 2001. The dataset is a blended product, using the global HadISST OI dataset prior to 1981 and the Smith/Reynolds EOF dataset post-1981 (see Hurrell, 2002). In addition to the analyzed time series, a composite of the annual cycle for the period 1981-2001 is also available in the form of a mean “climatological” dataset. The sea–surface temperature and sea ice concentrations are updated every time step by the model at each grid point using linear interpolation in time. The mid-month values have been evaluated in such a way that this linear time interpolation reproduces the mid-month values.

Earlier versions of the global atmospheric model (the CCM series) included a simple land-ocean-sea ice mask to define the underlying surface of the model. It is well known that fluxes of fresh water, heat, and momentum between the atmosphere and underlying surface are strongly affected by surface type. The CAM6.0 provides a much more accurate representation of flux exchanges from coastal boundaries, island regions, and ice edges by including a fractional specification for land, ice, and ocean. That is, the area occupied by these surface types is described as a fractional portion of the atmospheric grid box. This fractional specification provides a mechanism to account for flux differences due to sub-grid inhomogeneity of surface types.

In CAM6.0 each atmospheric grid box is partitioned into three surface types: land, sea ice, and ocean. Land fraction is assigned at model initialization and is considered fixed throughout the model run. Ice concentration data is provided by the external time varying dataset described above, with new values determined by linear interpolation at the beginning of every time-step. Any remaining fraction of a grid box not already partitioned into land or ice is regarded as ocean.

Surface fluxes are then calculated separately for each surface type, weighted by the appropriate fractional area, and then summed to provide a mean value for a grid box:

{F_{\psi}}_T = a_i \, {F_{\psi}}_i + a_o \, {F_{\psi}}_o + a_l \, {F_{\psi}}_l \, \;,

where F denotes the surface flux of the arbitrary scalar quantity \psi, a denotes fractional area, and the subscripts T, i, o, and l respectively denote the total, ice, ocean, and land components of the fluxes. For each time-step the aggregated grid box fluxes are passed to the atmosphere and all flux arrays which have been used for the accumulations are reset to zero in preparation for the next time-step. The fractional land values for CAM6.0 were calculated from Navy 10-Min Global Elevation Data. An area preserving binning algorithm was used to interpolate from the high-resolution Navy dataset to standard model resolutions.

The radiation parameterization requires monthly mean ozone volume mixing ratios to be specified as a function of the latitude grid, 23 vertical pressure levels, and time. The ozone path lengths are evaluated from the mixing–ratio data. The path lengths are interpolated to the model \eta–layer interfaces for use in the radiation calculation. As with the sea–surface temperatures, the seasonal version assigns the monthly averages to the mid–month date and updates them every 12 hours via linear interpolation. The actual mixing ratios used in the standard version were derived by [Che86] from analysis of [Dutsch86].

The sub-grid scale standard deviation of surface orography is specified in the following manner. The variance is first evaluated from the global Navy 10^\prime topographic height data over an intermediate grid (2^{\circ} \times 2^{\circ} grid for T42 and lower resolutions, 1.67^{\circ} \times 1.67^{\circ} for T63, and 1.0^\circ \times 1.0^\circ for T106 resolution) and is assumed to be isotropic. Once computed on the appropriate grid, the standard deviations are binned to the CAM6.0 grid (i.e., all values whose latitude and longitude centers fall within each grid box are averaged together). Finally, the standard deviation is smoothed twice with a 1–2–1 spatial filter. Values over ocean are set to zero.