Mesoscale activity in the North Sea as seen in ensemble simulations

Abstract

A set of two simulation ensembles of the ocean circulation in the North Sea, the Skagerrak and bordering seas has been run for the ten year period that started in January 1992. The ensembles differed only in the horizontal grid resolution. The main purposes of this investigation are (1) to quantify the variability that can be expected in multi-year simulations due to noise-like perturbations in the initial fields, and (2) to examine the robustness of model results for mesoscale features that form on the front between the Norwegian Coastal Current and water masses that are of an Atlantic Ocean origin. It is shown that the model resolution has a substantial impact on the ensemble variability, and that the role of small perturbations become more significant as the grid mesh is refined. Nevertheless, it is demonstrated that in a region to the west of the southern tip of Norway, eddies are occasionally found in the same positions at the same time in the results from all members of the ensembles. This is particularly the case in the aftermath of outbreak events of low salinity water masses from the Skagerrak into the North Sea.

Appendix: Properties of ensemble variability

Since the ensemble members differ only in their initial states, differences in the results can be attributed to nondeterministic differences in both the initial conditions and the evolution of the simulations. A technique to separate variability of a prognostic variable into two components was suggested by Metzger and Hurlburt (2001), and also used by Melsom et al. (2003). The technique was recently refined by Melsom (2004), and is repeated here for the sake of completeness.

Consider results for a prognostic variable η at a point in space, and define a partitioning of η by

$$ \eta_{s}(n) = \tilde{\eta}_{s}(d_{n}) + \hat{\eta}(n) + \eta^{\prime}_{s}(n), $$

(3)

where s is a member of ensemble simulations. Here, n is the time step, and d _n is the decimal day of time step n. $\tilde{\eta}_{s}(d_{n})$ is the daily climatology for member s on day d _n . Further,

$$ \hat{\eta}(n) = \frac{1}{S} \sum\limits_{s=1}^{S}{[\eta_{s}(n) - \tilde{\eta}_{s}(d_{n})]} $$

(4)

is the mean offset from the daily climatologies. Then, from (Eq. 3) we see that η′ is the departure of each member from the instantaneous ensemble mean as a function of space and time so that

$$ \overline{\eta^{\prime}_{s}(n)}^{s} = \sum\limits_{s=1}^{S}{\eta^{\prime}_{s}(n)} = 0, $$

(5)

where the overbar corresponds to an average over the ensemble members.

Using Eqs. 3 and 5, the mean square offset from the daily climatology $(\tilde{\eta}_{s})$ may be expressed as

$$ \phi^{2} = \frac{1}{S} \sum\limits_{s=1}^{S}{[\eta_{s} - \tilde{\eta}_{s}]^2} = \hat{\eta}^{2} + \overline{{\eta^{\prime}_{s}}^{2}}^{s}. $$

(6)

Here, the final term on the right hand side is the instant local variance of the ensemble members. The first term on the right hand side is the local square offset of the ensemble mean from the daily climatology.

Keeping in mind that $\hat{\eta}$ is independent of the variability from one member to another, the temporal mean of $\hat{\eta}^{2}/\phi^{2}$ is an estimate of the fraction of deterministic variability in response to atmospheric forcing. The nondeterministic variability fraction which is due to flow instabilities, is then given by

$$ r^{\rm nd} = \overline{{\overline{{\eta^{\prime}_{s}}^{2}}^s}/{\phi}^{2}}^n .$$

(7)

Thus, r ^nd enables us to quantify the contribution from nondeterministic variability as a fraction between 0 and 1 at each grid node. Note that the accuracy of this estimate depends on the size of the ensemble.

In the present study the daily climatology d _n was based on model results from November 1996–October 2001, and a 30-day box filter was applied when the daily climatologies were computed. Decimal days were defined by numbering days from 1 January 1992 in cycles of 365.25 days.

Figure 11 displays the results for bSSH at a position in the eastern North Sea. The local daily climatology is given by the dashed line, the eight ensemble members are depicted by thin solid lines, and the ensemble mean is shown as the thick solid line. Then, the offset $\hat{\eta}$ of the ensemble mean from the daily climatology is the distance from the dashed line to the thick solid line. The offset η′ _s of ensemble member no.s from the ensemble mean is the distance from the thick solid line to the thin line that displays the results from this member. The corresponding terms on the right hand side of Eq. 6 are illustrated in Fig. 12, where the ensemble variance and the square offset from the daily climatology are depicted as functions of time by the thick and thin lines, respectively.

Fig. 11

Baroclinic sea surface height (in m) during September and October 2000. Results from the CRE are displayed for 3°50′E, 61°42′N, near the west coast of Norway. Thin lines correspond to the eight members, and the thick solid and dashed lines correspond to the ensemble mean values and the daily climatologies, respectively

Fig. 12

The ensemble variance (thick line) and the square offset from the daily climatology (thin line), based on the results in Fig. 11. Values along the vertical axes are in m². Note that the resolution along the vertical axes differ by a factor of 10