Let's start by taking a look at Fig. 8 from Lin and Arakawa 1997. At the bottom of the plot, near MSE value of 339 K, several lines emanate from a common point. This is the "plume base" which we can think of as the point near the surface where a parcel could begin to rise and follow different paths (the lines) depending on how much it mixes with the environment. These lines travel nearly straight up, but are a little bent, and some more spread to the left than others. These represent the MSE of cloud plumes with different rates of entrainment of environmental air. The MSE of the environmental air is the solid line that bulges out far to the left, and the dashed line (hard to notice except at the bottom right of the graph) is the environment's saturation MSE. It is the MSE the environment would have if it had extra moisture so its relative humidity would be 100% given its temperature. Note the cloud lines are all greater than this environment saturation MSE, hereafter MSE*. (At least this is true a little bit above cloud base the level of free convection LFC.) So that's why they are clouds.
Now take a look at my figure made today showing basically the same thing: two black lines for the environment average MSE and MSE*, as well as 4 different colored lines representing different parcel paths ending at different levels on the MSE* curve. Each colored curve represents a parcel entraining a constant amount of environmental air as it rises, that's why the MSE value is reduced from the starting value. Each curve ends at a different height as it touches the MSE* curve. Look at the top of the orange/yellowish line: if it were to continue on any farther to the left of the MSE* curve, this would make it negatively buoyant and unsaturated and would cease to be a cloud. That's why cloud top is assumed to be the NBL.
Now let us consider the green curve for a moment. It rises to the left of MSE* the entire time! This means a parcel following that path (entraining the specified constant amount with height) would always be negatively buoyant. Then it must not be a cloud at all, but a kind of mathematical fiction. The figure seems to imply that it is possible to have a cloud all through the atmosphere if it intends to stop at the top of the atmosphere, but impossible to have a cloud through to the mid-levels if it intends to stop at a mid-level. Does this mean the atmosphere is so unstable to convection there just couldn't be a cloud with a cloud top at mid-levels? No, it is an illustration of the limitations of a starting assumption. To understand why, take a look at this annotated version of the previous figure.
First look at the purple arrows. The top purple arrow points to the value of MSE* that the green curve plume is "aiming" for. Note that at this mid-level it is almost the minimum of MSE*. The purple arrow pointing down and the lower horizontal purple arrow show where the environment has the same value of MSE. Now consider the black arrows numbered 1 through 3. If the green curve were representing a parcel that was not mixing with the environment, it would rise straight up, not curve to the left. Black arrow 1 represents the "move to the left" the parcel MSE experiences by mixing with the environment value of MSE that the arrow is pointing to. We see that as we go up to arrow 2, the environment MSE continues to reduce, and mixing with it further reduces the MSE value of the green curve. However, the environment MSE is still greater than the MSE* at the mid-level cloud top we considered with purple arrows. It is not until the level of the bottom purple arrow that the parcel can even begin to mix with air with an MSE lower than the value it is aiming for. As illustrated by black arrow 3, this mixing allows the parcel's average MSE to fall to the MSE* it was aiming for.
Because we assumed a constant rate of entrainment with height, the parcel appears negatively buoyant throughout the path. As you may have already figured out, the way to get the green curve from the starting point to its ending point but have it be greater than MSE* for at least some of the path is to have it mix very little at first (rise more straight up than to the left) and then mix very much at higher levels and quickly move to the left, as illustrated by the green arrows. An entrainment rate starting very small and increasing exponentially with height would accomplish this. Indeed, an exponential entrainment rate is the solution to the entrainment relation laid out in Arakawa Schubert 1974. The RAS scheme makes the assumption that the entrainment rate is linear to reduce computational complexity.
Here we have shown that for the Giga-LES (GATE) case, a non-linear entrainment rate assumption can be critically necessary for properly simulating low and mid-level convective plumes.
No comments:
Post a Comment