I'm confused about the effect of gravity anomalies on sea height

[Jaams]

TL;DR

Lower than average weight at the equator causes the earth to be wider, yet positive gravity anomalies are claimed to cause an increase in sea height.

Summary: Lower than average weight at the equator causes the Earth to be wider, yet positive gravity anomalies are claimed to cause an increase in sea height.

Summary: Lower than average weight at the equator causes the Earth to be wider, yet positive gravity anomalies are claimed to cause an increase in sea height.

I'm trying to understand how differences in gravity would in practice end up affecting the local height of the sea. We know that at the equator, everything weighs less, and the sea-level is also higher up than average, seeing as the Earth's wider at the equator. But then if you look at a gravity map, you can see that they say that places with stronger gravity will have a sea-level that's higher up. That's in my opinion a contradiction, and I'm having trouble figuring out what I'm missing.

I'll attach the gravity maps made based on sea-level height, then mGal, and then a picture of the Earth's oblate shape caused by Earth's rotation:

The images above are sadly oriented differently, but the blue and red areas correspond with each other, showing that increased gravity = increased sea-level.

Here you can see the diameter of the Earth and how it increases as you get closer to the equator.

I'm hoping that someone can explain to me how this works. If anyone knows of any website or earlier thread that goes over this, then that's good too. I wasn't able to find one.

 

[Dale]

Notice the scale of these plots. The first one is in meters and over the Pacific Ocean it ranges from about -50 m to about +10 m with little or no prolateness in evidence. The globe is in km and ranges from about 6361 km to about 6373 km over the Pacific Ocean with strong prolateness.

Basically, the first plot has subtracted off the prolateness of the second so that you can see the small little deviations that you cannot see on the larger scale.

 

[Andrew Mason]

The seas will tend to form in such a way to create an equipotential surface. In other words, where the gravitational potential is lower, water will necessarily flow in so that the water across the sea surface has equal potential energy. Where there is more local gravity, the seas will have lower potential energy at the same surface height. So water from surrounding areas will flow so increase the sea surface height to that the gravitational potential of water at the surface is equal across the entire surface. This only works with a fluid such as water.

Centrifugal effects due to the Earth's rotation also affects gravitational potential. This effectively lowers the rate of gravitational free-fall at a given distance from the Earth centre at the equator compared to locations at higher northern or southern latitudes. So water at the surface will tend to flow away (slightly) from the equator toward the poles to create an equipotential surface. This has to be factored in when trying to map local gravity effects due to mass variations under the seas.

AM

 

[Jaams]

Dale, you're right, but my confusion lies with a different issue and I'll explain it below:

Andrew Mason, to summarize (based on my interpretation), you're saying that the positive gravity anomalies draw more water in and the sea level rises there, and that water is drawn to the poles from the equator for the same reason.

I'm confused by that, because water would in my opinion need to be drawn to the equator for the Earth to be oblate, and for there to be any water at the taller equator in the first place.

I hope this response makes my source of confusion more apparent.

 

[A.T.]

I hope this response makes my source of confusion more apparent.

Use vectors instead trying to relate local gravity to water level.

 

[Jaams]

Use vectors instead trying to relate local gravity to water level.

I don't know how you want me to actually use vectors and approach this, but I'll try.

I'm aware of how gravity and the centrifugal force add up to an oblate shape, and I get that if there were spots of high local gravity added into that calculation, at those locations level would be more towards the center than in surrounding ones, suggesting that a high gravity spot will decrease sea height.

But that doesn't really take into account much at all (I only took direction into account), so I have no confidence in that.

 

[A.T.]

... suggesting that high gravity spot will decrease sea height...

As I said, don't try to relate the local strength of gravity to the water level. You have to look at the effective potential instead. The water will form an equipotential surface.

 

[Andrew Mason]

I'm confused by that, because water would in my opinion need to be drawn to the equator for the Earth to be oblate, and for there to be any water at the taller equator in the first place.

I hope this response makes my source of confusion more apparent.

We think of water flowing downhill, but that is because "downhill" is toward a region of lower potential. Where it gets a bit complicated is with the effect of the Earth rotation.

It seems a bit counter-intuitive but being closer to a large mass reduces gravitational potential. That is because potential at a particular position is the energy that must be added to a unit mass at that position in order for it to escape the gravitational field entirely. It is always a negative quantity:

$U(r) = - G\int \frac{dM}{r_i}$ where $r_i$ is the distance from the position of the unit mass (at radial distance r to the centre of mass of M) to the gravitating mass element dM .

So water at the same distance from the Earth centre but not near a large positive gravity anomaly will tend to flow toward that anomally which will build up a higher sea level there.

Factoring in the effect of the Earth's rotation is a bit trickier. I stand to be corrected, but here is my take on it:

The Earth crust (water excluded) is rigidly held together - it can't flow. It is kept together with gravity. Pressure from the weight of the Earth "above" the centre of mass (crust, mantle, iron core etc.) bears on the centre of mass and keeps everything compacted together. If the weight of the Earth was completely uniform in all directions there would be uniform pressure down to the core and the Earth shape would approach a perfect sphere. But the weight is not uniform because the Earth is rotating. The centrifugal effect (perceived as a force in the rotating reference frame of the earth) reduces that weight, having maximum effect at the equator and decreasing to zero at the poles. This pressure difference causes the Earth to bulge slightly at the equator. The reduced weight of a unit mass at the equator a distance r from the Earth centre of mass means it has a higher gravitational potential compared to an identical unit mass at the pole but at the same distance from the centre of mass of the Earth (ie. the rotation reduces slightly the energy required for the unit mass to escape the earth).

Since water, as a fluid, will seek a region of lower potential, sea height would tend to be lower near the equator, everything else being equal. Otherwise there would be a higher gravitational potential at the surface of the ocean at the equator than at the surface a distance away from the equator, in which case water would flow away from the equator to the lower potential surface.

AM