Vertical Treatment#

The external reference data used for nudging may be supplied on its own set of hybrid-pressure levels, which do not in general coincide with the model’s levels. Before the reference fields can be used in the nudging increment, they must therefore be interpolated onto the model’s levels, and the temperature field converted to potential temperature. A height-dependent weight is also computed, to allow nudging to be restricted to a chosen range of vertical levels.

Reference pressure profile#

The reference dataset’s hybrid-pressure levels are defined by coefficients \(a_k\), \(b_k\). Given the reference surface pressure, \(p_{s,ref}\), the pressure on reference half-level \(k\) is

(220)#\[p_{ref,k} = a_k + b_k \, p_{s,ref} ,\]

and full-level values are formed by averaging \(a_k\) and \(b_k\) between adjacent half levels.

Model pressure profile#

The corresponding pressure on the model’s levels is obtained from the model’s Exner pressure, \(\Pi\), evaluated on both sets of model levels (those on which pressure is held, and those on which potential temperature is held):

(221)#\[p = p_0 \, \Pi^{1/\kappa}\]

where \(p_0\) is a reference pressure and \(\kappa = R_d / c_p\), with \(R_d\) the gas constant for dry air and \(c_p\) the specific heat capacity at constant pressure.

Temperature to potential temperature#

The reference dataset supplies temperature, \(T_{ref}\), rather than potential temperature. This is converted using the reference pressure profile computed above:

(222)#\[\theta_{ref} = T_{ref} \left( \frac{p_0}{p_{ref}} \right)^{\kappa} .\]

Vertical interpolation#

The reference fields (\(\theta_{ref}\), \(u_{ref}\), \(v_{ref}\)) are interpolated from the reference pressure levels onto the model’s pressure levels by linear interpolation in log-pressure:

(223)#\[X_{ref}(p) = X_{ref,l} + \left( X_{ref,l+1} - X_{ref,l} \right) \frac{\ln p - \ln p_{ref,l}}{\ln p_{ref,l+1} - \ln p_{ref,l}}\]

where \(l\) is the reference level immediately below the model level being interpolated to (in pressure). To avoid unbounded extrapolation above the top or below the bottom of the reference data, the interpolation factor is limited to the range \([-1, 2]\). The lowest model \(\theta\) level is set equal to the value interpolated for the level immediately above it.

Vertical tapering#

To allow nudging to be targeted at a chosen range of vertical levels, a height-dependent weight, \(w_{taper}\) (introduced in (209)), is computed for each function space (\(\mathbb{W}_3\) and \(\mathbb{W}_\theta\)). The weight ramps linearly from 0 to 1 between a lower taper level and a lower full-strength level, remains at 1 between the full-strength levels, and ramps back down to 0 between an upper full-strength level and an upper taper level:

(224)#\[\begin{split}w_{taper}(\tilde{k}) = \begin{cases} 0 & \tilde{k} \le k_{taper,bot} \\ \dfrac{\tilde{k} - k_{taper,bot}}{k_{bot} - k_{taper,bot}} & k_{taper,bot} < \tilde{k} < k_{bot} \\ 1 & k_{bot} \le \tilde{k} \le k_{top} \\ 1 - \dfrac{\tilde{k} - k_{top}}{k_{taper,top} - k_{top}} & k_{top} < \tilde{k} < k_{taper,top} \\ 0 & \tilde{k} \ge k_{taper,top} \end{cases}\end{split}\]

where \(k_{bot}\) and \(k_{top}\) are the model levels marking the bottom and top of the region over which nudging is applied at full strength, and \(k_{taper,bot}\) and \(k_{taper,top}\) are the levels below and above which nudging is zero. The quantity \(\tilde{k}\) is the \(\mathbb{W}_\theta\) level index starting from 0, while \(\mathbb{W}_3\) points are offset by half a level. Using this common coordinate ensures that both sets of levels see a consistent, vertically-aligned taper profile.

Tropopause capping#

Nudging towards external reference data is generally undesirable well above the tropopause: the stratosphere is only loosely constrained (if at all) by the coarse vertical and temporal sampling of typical reference datasets, and imposing large-scale forcing there can disrupt the model’s own stratospheric transport and composition. Rather than relying solely on a fixed upper taper level, chosen in advance to sit safely below the tropopause for an expected range of conditions, the upper extent of the tapering profile described above may instead be capped dynamically by the model’s own, evolving tropopause height.

When this option is used, the upper full-strength and taper levels, \(k_{top}\) and \(k_{taper,top}\), in (224) are replaced at each application of the scheme by the lesser of their configured values and the current tropopause level, diagnosed independently for each atmospheric column from the model’s evolving temperature structure. This allows the vertical extent of nudging to track a rising or falling tropopause, rather than remaining fixed, and ensures nudging is switched off in the stratosphere even where the diagnosed tropopause sits below the configured upper taper level. A fixed minimum level may also be enforced as a floor for the diagnosed tropopause level, guarding against spuriously low diagnoses; setting this floor high enough disables the dynamic capping altogether, recovering the fixed taper profile described above.