Yeah, my bad, I misunderstood. Actually I was surprised that the Earth would be within the Hill Sphere of the red dwarf in that case, but it is and the scenario works. It is quite interesting!So basically we'd be adding an additional 22.8% solar flux to the Earth, which would raise the average temperature by about 10 to 15 degrees C, (before accounting for climate feedbacks, which would probably push it higher), and this roughly agrees with what Jack shows in SE.
A ~10c warmer world is difficult to contemplate, and we have to look back at least as far as the Paleocene-Eocene Thermal Maximum (~50 million years ago) to find a comparable climate state in Earth history. During the PETM there was a mass migration of species as temperatures rose, and subtropical species even made it to the poles. Species that could not adapt or migrate died out, particularly bottom ocean dwellers. Weather patterns similarly shifted poleward, and precipitation was much more intense with the extra water vapor in the atmosphere.
In this case though, Earth's rotation also changes from a day to a month, and the heat would be distributed differently. About 1/5 of the heat lands on a fixed side of Earth, while the other 4/5 rotates around the Earth with the "solar day", which lasts a month (29.5 days, assuming Earth orbits prograde). This is a pretty wonky climate situation. When the red dwarf is near its "new phase", the side of Earth facing it would be hottest, while the other side is in cold night. When near "full phase" the heating is more even, but the side facing the red dwarf is colder. So there would be a monthly cycle in the atmospheric circulation and weather patterns, with a "hot pole" that rotates around the planet with the sun and is strongest when it is aligned with the red dwarf.
I'm not sure if it would get cold enough for oceans to start freezing on the night (sun-relative) side. I'm inclined to doubt it because this oscillation is pretty short (not a whole lot of time to heat up and cool down each cycle) and the oceans and atmosphere are pretty effective at transferring the heat. Basically I imagine a flow of wind and storms away from the sub-solar point, just like a typical tidally locked planet, except that this flow also moves and changes intensity.Most species would be in trouble if Earth suddenly found itself in this scenario, but if the system evolved this way then life could adapt to it and probably even thrive. Even photosynthesis should be fine with the seasonal loss of sunlight -- it already manages this in the polar winters which last a lot longer.
I would really love to see a global climate model simulating this.