Rivers that flow "uphill" due to Earth's rotation

Question

One of the fun tidbits that gets bandied around in physics departments is that the Mississippi River actually flows away from the center of the Earth. Due to the Earth's oblateness, the source of the Mississippi is actually significantly closer to the Earth's center than the mouth is, and it's only because of the Earth's rotation (and the centrifugal force present in a reference frame rotating with the Earth) that the water in the Mississippi ends up in the Gulf of Mexico.

My questions are two:

1. Does this mean that the Mississippi would flow north if the rotation of the Earth stopped? I'm leery of just looking at the distance to the center of the Earth, since the oblateness of the Earth also causes a distortion of its gravity field (i.e., the equipotential surfaces are no longer perfect spheres), and this distortion would seem to be on the same order of magnitude as the small differences in elevation we're considering. In other words, if one uses the actual gravitational potential for a non-rotating oblate spheroid, is the Gulf of Mexico at a higher gravitational potential than Minnesota?

2. Are there any other rivers in the world for which this is true? Obviously these would need to be rivers that flow towards the equator with a very shallow gradient (elevations measured relative to "sea level", which would be one of the equipotentials measured above.)

EDIT: For clarity's sake, assume that the Earth would remain the same shape if it wasn't rotating. (In reality, hydrostatic forces would eventually pull a non-rotating Earth into sphericity, but for this question I'm interested in the difference between an oblate, rotating Earth and an oblate, non-rotating Earth.)

Answer 1

2. Are there any other rivers in the world for which this is true?

The Mekong, at least after it has left the Tibetan plateau.

Location          Latitude       Elevation (m)  Radial distance (km)
Source            33° 42' 30"       5224             6376.8
Manwan Reservoir  24° 45' 15.5"      997             6375.4
Ruak River mouth  20° 21' 16"        336             6375.9
Mekong delta       9° 27' 30"          0             6377.6

While the Mekong does flow "downhill" (toward the center of the Earth) at the start thanks to that four kilometer drop in altitude across the Tibetan plateau, it's pretty much all uphill from the Manwan Reservoir on.

1. If one uses the actual gravitational potential for a non-rotating oblate spheroid, is the Gulf of Mexico at a higher gravitational potential than Minnesota?

There's a nice what-if scenario at esri, the Environmental Systems Research Institute, that asks and answers this question: If the Earth Stood Still: Modeling the absence of centrifugal force. The scenario investigates what would happen to the Earth's waters if the Earth somehow stopped rotating over the course of a few decades. That's a long enough span of time that the waters would have a chance to adjust, but far too short a span of time for isostasy to readjust the shape of the Earth.

Not only would the Mississippi stop flowing southward, the waters of the oceans would flow away from the equator. This would leave a globe-spanning equatorial bulge of land surrounded by two polar oceans. The end result:

_{Source: https://www.esri.com/news/arcuser/0610/nospin.html}

Answer 2

I can give a brief answer, though if anyone wants to do a better, more detailed one, feel free.

Sea level at the Equator is 21.36 km higher than at the poles. Source, and with a distance of about 10,000 km pole to equator, that's a natural rise moving towards the equator of 2.1 meters per km. That works out to 2.136 meters per km or a bit over 11 feet per mile, a 1 mile drop every 468 miles and I'm not sure earth's bulge spreads consistently, but a 1 mile drop over 468 miles is quite a bit and likely more than most longitudinal elevation drops. (I'm not sure longitudinal elevation drops is a proper term, but I like the way it sounds).

I'd wager that most equator bound rivers flow "up hill" compared to the center of the earth over most of their routes. I looked, but couldn't find any good lists for river elevation drops so I couldn't verify.

Does this mean that the Mississippi would flow north if the rotation of the Earth stopped?

Pretty much yes, but if the Earth stopped (and you ignore the other effects of stopping a 1,000 MPH rotation) then, not just the Mississippi but the Oceans and all water would flow North and not in small amounts. All of Alaska might be submerged and most of Canada and the equator, even around the oceans might lose all its water and form a land bridge. A 21.36 mile bulge in sea level is a LOT and if that suddenly goes away. Wow. The effect would extend way way way way past the Mississippi. This effect, of course, would only be temporary as the land too would correct itself from the Equatorial bulge too, just more slowly than the oceans and you might see a number of 11 or 12 or even 13 Richter scale earthquakes as the Earth made gravitational adjustments to undo its rotational bulge. (I rather like what if scenarios like this one. Kind of fun to imagine what they would be like).

since the oblateness of the Earth also causes a distortion of its gravity field (i.e., the equipotential surfaces are no longer perfect spheres), and this distortion would seem to be on the same order of magnitude as the small differences in elevation we're considering. In other words, if one uses the actual gravitational potential for a non-rotating oblate spheroid, is the Gulf of Mexico at a higher gravitational potential than Minnesota?

The Gulf of Mexico is actually lower gravity than Minnesota, for two reasons. The rotation effect, which you mention is bigger as you move towards the equator and that's the larger of the 2 effects but being further away from the center of mass also reduces gravity. You might think because there's more land under you that you'd weigh more, but you're also further from the center of mass, so the effect is that, even when you factor out rotation, you weigh slightly less in the Gulf of Mexico than in Minnesota, just as you weigh less at the top of Mt. Everest than at the foot of Everest. explained here