The Mighty Mississippi River has been in the news lately. In short, the river level is pretty darn low causing navigation difficulties. Boats are running into the bottom of the river due to the lack of depth.

Question: How much rain does the Mississippi river need to maintain its water level?

Great question. Now, here let me make a point. The point is not to get the answer to this question. Rather, the point is enjoy the process of estimating an answer like this. I suspect that you could search for an answer online. Maybe one already exists. However, letting the existence of an answer stop me from doing an estimate would be like not climbing Mt. Everest since someone else already did it.

Estimating Flow Rate ——————–

How much water comes out of the Mississippi? Let me guess. If I look at the mouth of the river, I could assume the following (with my estimates):

River width: w = 1000 m.

River depth: d = 6 m.

Water speed: v = 1.5 m/s.

Yes. I know that these values could be way off. I am just guessing here. Now let me assume a rectangular cross section to the river and a constant water speed (it actually travels faster in the middle). In a certain amount of time (Δt), how much water will move through this cross sectional area?

The volume of this water would be:

Of course, I don't really care about this volume of water. I care about the flow rate - or the volume of water per unit of time. Let me call this flow rate, f.

Since the two length variable have a unit of meters and the speed is in m/s, this means the flow rate is in m3/s. If I put in my guesses from above, I get a value of 9,000 m3/s. That seems a little low for the Mississippi, but I will proceed.

Rain Rate ———

Maybe my flow rate is off, but I still think that all of this water has to come from rain and snow (well, most of it). If you want a stable system, water in equals water out. Actually, the case of the Mississippi there is probably more water output than what I calculated. Where does the rest of the water go? Lots of people use the water from the river for things like farming and industry. Ok, but I am going to go with the assumption that all of this water coming out the mouth of the Mississippi comes from rain (and snow) falling in the area of land that drains through the Mississippi.

Image modified from National Park Services

How big is the Mississippi watershed? Of course there are more sophisticated methods of determining the size of this area, but I went with a simple approach. I mean, I am estimating the flow rate so why be exact on the watershed size? For this area, I have a rectangle that is 1.5 million meters by 2.2 million meters. This gives an area of 3.3 x 1012 m2.

The common method for recording rainfall is in inches. If I want the flow rate over this larger area to be the same as the flow rate of the mouth of the Mississippi, how many inches of rain would that be in a month? First, what is the volume of water from the Mississippi in 1 month? If I use a Δt of 2.63 x 106 seconds (same as 1 month), then the volume of water would be:

Now, if this volume of water was rain spread over the whole watershed, the depth of this rain would be:

Using my estimate for the area of the watershed, I get 0.0072 meters or 0.28 inches of rain per month. Really you would need more rain than this. This is the rain that drains (or the drain rain). Rain probably does several things when it falls. I suspect that probably half of the rain fall evaporates back into the air before it gets down the Mississippi. Although the ground absorption would probably have an overall neutral impact on the amount of rain. Another point is that this calculation is the average rainfall over the whole watershed. I'm sure there are some parts of the watershed that get much less than 0.28 inches and some parts that get more.

One more thing. I thought I would check my estimates. The Wikipedia page on the Mississippi river lists an average flow rate of 16,800 m3/s. So, I was off by more than a factor of 2. Not a big deal - at least I wasn't off by a factor of 100. If I use the Wikipedia value for flow rate, there would need to be 0.52 inches of rain per month in the watershed. I can believe that.

How much rain do you get on a rainy day? This is something that a lot of people don't have a great intuitive feel about. If you need to get 1 inch of rain over a month (to account for evaporation and stuff), what would that be like? Let's look at the rain over the past month for my location. Weatherspark.com is a great site for getting weather data in a nice graphical format.

This plot (from Weatherspark) shows the precipitation over the past month.

You can see that for a couple days at the end of November, there was some rain. It was only about 0.02 inches or less. At that rain rate, it would just about have to rain every day of the month to get up to the 0.52 inches. However, look at the rain in December. We had some serious rain with one day alone over 0.5 inches. I will tell you, that was some hard rain and not very common.