Formulation#

Let \(X\) denote a field being nudged (potential temperature \(\theta\), or one of the horizontal wind components \(u\), \(v\)), and let \(X_{ref}\) denote the corresponding reference field, having already been vertically interpolated onto the model’s levels (see Vertical Treatment). At each call to the nudging scheme, an increment \(\Delta X\) is calculated and added to the model’s tendency for \(X\).

Two methods are available for calculating \(\Delta X\).

Newtonian relaxation#

In Newtonian relaxation, the increment is simply proportional to the local difference between the field and its reference value:

(207)#\[\Delta X = w \left( X_{ref} - X \right)\]

where \(w\) is a weight described in Nudging weights below. This relaxes \(X\) directly towards \(X_{ref}\) at every horizontal scale present in the reference data, with no spectral selectivity.

Convolution#

In the convolution method, the local difference between the field and its reference value is first filtered so that only the large scales are retained, before being scaled by the same weight \(w\):

(208)#\[\Delta X = w \, \mathcal{L}\left[ X_{ref} - X \right]\]

where \(\mathcal{L}\) is a scale-selective filter that removes the short wavelengths. Rather than performing this filtering with a global spectral transform, \(\mathcal{L}\) is applied as a convolution in physical space. The motivation for this choice, and the derivation of the convolution kernel on the sphere, are described in Filtering by Convolution.

Nudging weights#

In both of the above methods, the increment is scaled by a single weight \(w\), which combines a vertical tapering factor, a spin-up ramp, and a relaxation-timescale factor:

(209)#\[w = w_{taper} \; r_{spinup} \; \min\left(1, \frac{\Delta t}{\tau}\right)\]

where:

  • \(w_{taper}\) is a height-dependent factor, computed as described in Vertical Treatment, that ramps the nudging on and off over a chosen range of vertical levels;

  • \(r_{spinup}\) is a factor that ramps linearly from 0 to 1 over a configured spin-up period, described below;

  • \(\Delta t\) is the model timestep and \(\tau\) is a configured relaxation timescale. If \(\tau\) is set to zero, this factor is instead set to 1, so that the field is relaxed fully towards the reference state within a single timestep.

Spin-up ramp#

The nudging weight can be ramped up gradually over a configurable spin-up period \([t_{start}, t_{end}]\):, allowing forecasts over short time scales to be unperturbed.

(210)#\[\begin{split}r_{spinup} = \begin{cases} 0 & t \le t_{start} \\ \dfrac{t - t_{start}}{t_{end} - t_{start}} & t_{start} < t < t_{end} \\ 1 & t \ge t_{end} \end{cases}\end{split}\]

where \(t\) is the elapsed model time since the start of the run. If \(t_{start}\) and \(t_{end}\) are equal, the ramp is skipped and \(r_{spinup} = 1\) once \(t\) exceeds this value.