A weak (and slowly further weakening) Atlantic low will linger near or over Ireland most of this week. You may have heard a forecast for "cyclonic winds" and that refers to the fact that with the low situated at about the latitude of north Connacht, some northern areas will have strong east winds (Malin Head is showing that at present with gusts to about 85 km/hr there). Some areas further south will have winds from various directions controlled by the low pressure, for example, the east coast can expect south to southwest winds, the west coast more like west to northwest especially later today and tomorrow when the low finally moves onto land. So the term cyclonic here refers to the circular counter-clockwise wind flow always found near low pressure areas in this hemisphere.
Our current example will continue to push slowly east and then north for a time (Thursday into early Friday) before being pushed back to the south later Friday into Saturday.
That's going to allow a very gradual cooling trend as the weak low becomes encircled by somewhat cooler air from the northeast, not the full arctic air mass that might arrive next week though. Temperatures each day this week will likely be about one degree cooler each day, except that in north Ulster they will rise as the easterly is temporarily cut off. So expect highs of about 8-10 C today, 7-9 C Thursday, 6-8 Friday and probably Saturday also, then perhaps as low as 4-6 C by Sunday when the influence of the low should be replaced by the first stages of the (potentially) developing cold easterly.
In that interval, overnight lows will only be a few degrees lower than the highs due to a fair amount of cloud in the circulation, so would expect lows near 4 C tonight, and again dropping off very gradually to around 1-2 C by the weekend.
Showers or more prolonged intervals of rain with this dying low will be mostly confined to areas north of a Galway to Dublin line but not always so, and amounts will be fairly moderate (10 mm at most today, then 5-10 mm amounts for a few places each day after that, most places considerably less).
These showers will begin to become more wintry by Saturday and especially Sunday, as the flow turns around more to the northeast with the departure of the low towards southern Britain and then northeast France. At the same time, the fresher northeast breezes will likely mean a decrease in cloudiness for western counties which may get to see longer bright spells each day by the weekend.
The big question is, what happens next week? The model guidance has come together somewhat on that important question, but there is still a range of outcomes. The range has narrowed down to this essentially -- there almost certainly will be an easterly flow for several days, it will be at least somewhat colder than average for this time of year, and could be very cold indeed. The coldest days where guidance brings in the very cold option appear to be around Monday to Wednesday with some tendency on guidance for the cold spell to end gradually later in the week.
If it's the modified cold option that wins (currently strongest on the U.S. model) then there wouldn't be much snow potential and temperatures would level off around 3-4 C daytime and -3 C overnight. Even with that some isolated snow showers might develop near the east coast. If the considerably colder options prevail, and most other model guidance suggests that now, temperatures may drop as low as -2 to zero C daytimes, and -9 to -5 C for overnight lows. As in normally the case with these cold easterly spells, colder temperatures are found further west especially in Connacht, away from the brief warming influence of the Irish Sea, which, like the North Sea, gets a period of time to work on warming up the surface layers of these cold air masses. That process leads to the formation of unstable shower bands known as "streamers" which then infiltrate both eastern Britain and eastern Ireland in turn, dropping snow or perhaps mixed wintry showers if the cold isn't quite robust enough. In general, these bands will be all snowfall if the temperatures at about 1500 metres (above the height of all terrain in Britain or Ireland) fall below about -8 C, which usually guarantees surface readings around 2 or 3 C. When we have more marginal "uppers" then the tendency is for the streamers to produce a wider mix of showers. That could be the case at first here, on Sunday when the coldest air is starting to arrive.
Another mechanism for snowfall in these cold spells is the passage of low pressure areas to the south, without totally removing the surface layers of cold air. In those cases, the 1500m temperatures can be a bit more moderate to permit snow (-5 or -6 can work), since there is less of a maritime influence to consider, the cold air already being in place over Ireland when the moisture from the passing low is thrown over top of it. We can see several opportunities for such snowfalls in this coming cold spell, most notably towards the middle to end of next week. However, models have a fairly poor track record of "nailing down" these details at this time interval (currently about 7-10 days), so all I can say here is that in addition to the more reliable "streamer" type snowfalls in Leinster and parts of Munster and Ulster, there would be less predictable but more widespread snowfalls from passing lows.
Of course, any arrival of lows that close runs the risk of pushing too far north to sustain the snow very long before it then turns over to sleet and rain. That development is more supported by the milder GFS model that I mentioned, as being the outcome fairly quickly after the modified cold that it predicts has come and gone. However some guidance that allows robust cold also goes into this faster transition to milder Atlantic air masses by about Friday of next week.
So there's the overview and you can assess your forecast probability from that, I could pick a specific set of options and say that is what will happen, and in fact that's always what forecasters are doing, navigating what they think is the path of least resistance through a wide variety of outcomes (normally the first 2-3 days are fairly certain nowadays thanks to improvements in the technology of the models, after that, it becomes increasingly an educated guessing game based in large part on what confidence we have in different models, either against each other or within the range of their own internal uncertainties.) The GFS model helpfully (or perhaps not) provides an ensemble of about thirty solutions that could happen based on changing a few details of how the model runs, and this time out there is a very wide range in those outcomes after day five, which simply says the model recognizes that other model solutions could very well be correct, or at the other extreme, there might be a very brief and very modified cold spell. As always, time will tell but the bottom line is, the more significant cold and snow potential looks to be around a 70% chance now (applied to at least Monday to Wednesday if not a longer interval like Sunday to ??? ).
In the past, cold spells have locked in for weeks in February, but most of the examples are well back in the past. The first half of February 1991 was quite cold then it warmed up to a more normal level. In 1986 there was more sustained cold most of the month. Other notable past Februaries with long cold spells were (going back in time) in 1963, 1947, 1917, 1895 and 1855. Most of these years produced some significant snowstorms in eastern Ireland. So that's the pedigree of severe cold at this time of year, and in almost all these cases, there was simultaneously some robust cold outbreaks in eastern North America. That's interesting because this is being shown on a lot of guidance at the present time also.