I don't have a deep understanding of the climate on earth, i probably still can contribute insights what would happen from a place of high school physics.
The introduction of this energy sink is equivalent to moving the earth farther from the sun in terms of energy balance.
Looking at the earth at it is in terms of energy balance the earth should be at reflectivity of the surface it has, the distance of the sun, ... the earth should have an average temperature of -10 to -20 °C. The reason why it isn't that cold is the green house effect, the light spectrum earth emits back into space has different wave lengths which the atmosphere partially reflects back to earth.
The world climate has several different stable states it switch between. One of them is the ice age, one them is the preindustrial climate and one them is a hot house earth (which we had ~100 million years ago).
Each of these state is quiet stable to small changes and is able to adjust changes via stabilizing feedback mechanisms. This video by Nasa explains the Gaia theory which is relevant to the question.
However the introduction if such a 0°C wall is the kind of change which exceeds the capability of any stabilizing mechanism. Here is a rough time line what might happen:
- On the day and night side all points visible from the wall start to freeze over and snow falls in the equatorial region. As the earth turns that snow turns away but more and more snow accumulates faster than it melts.
- The snow in the best lighted region of earth changes the earths reflectivity (albedo) so that the earth absorbs less energy, this causes a global drop in temperature
- The oceans are largely unaffected visibly since sea water doesn't freeze at 0°C, however there is a giant change in global circulations, i can't predict.
- Some regions around 45° North / 45° South degree will become the new tropical zone. By that i mean a zone of highest temperature which serves as place where the hot air circulates with the cold air from north and south (only that one of these cold zones now is the equator).
These are the obvious first and second order effects, however i see 3 possible paths from here:
The previous moderated region becomes a becomes subpolar but stabilizes and land life can continue as is, although their numbers are decimated a small number persists there are two spheres of live which develop independently from now on, although flight between them is still possible. Life grows between glaciers and if intelligent live persists it tries to balance emitting green house gases and surviving another year. The ice might slowly recede and when they are back to the industrial age they are blessed with an infinite heat sink.
All land life gets crushed between glaciers and the earth is in a long maybe never ending ice age. The polar ice cap will expand farther but marine live will persist. Recovery might be possible if the magic ever ceases to exist. Life will be scared for and take millions of years to develop a new class of intelligent land animals.
The albedo feedback turned the complete planet in to an ice ball, this had lasting effects on the atmosphere which lost some of it's climate property, depending on how far away the equivalent ice wall less earths orbit is, some gases might have turned fluid. However life persists in the most unlikely place. Ocean water can't freeze at 0°C. So there are two thin lines of blooming with live next to the magic wall of life, slowly evolving, maybe changing the atmosphere. It is possible that they manage to flourish or expand there living area. Should the magic wall, now serving as heater ever vanish life on earth would be gone in an instant.
The question is not as ridiculous and could be answered properly using simulation. One would basically run a modified energy balance model, with the boundary conditions that the equator has 0°C. While this question is not possible under known physics. Physics still has an answer to this "What if".
If you want to simulate that here are some pointers:
An energy balance model but when the impact on the atmospheric convection and precipitation distribution wants to be modeled to you need a global climate model.
ISCA might be a code base you could modify. However an model particularly designed to answer question about Ice ages might also be a good start.
Anyways, that's a few directions. I'd look at both the directions of Esther Widiasih and the Minnesota group (+ Hans Kopper), and Dorian Abbot's GCM investigations of the topic, and pick something in that area.