# Isaac Held's Blog

Animation of near-surface wind speeds in rotating radiative-convection equilibrium, following Zhou et al, 2014.

I have discussed models of non-rotating radiative-convective equilibrium (RCE) in previous posts.  Given an atmospheric model one idealizes it by throwing out the spherical geometry, land-ocean configuration and rotation, creating a doubly-periodic planar geometry re-entrant in both x and y, while also removing any horizontal inhomogeneities in the forcing and boundary conditions.  In the simplest case, surface temperatures are specified and the surface is assumed to be water-saturated. The result is an interesting idealized explicitly fluid dynamical system for studying how the climate — especially that of the tropical atmosphere — is maintained by a balance between destabilization through radiative fluxes and stabilization through turbulent moist convection.  There is a lot that we don’t understand about this setup, which still contains all of the complexity of latent heat release and cloud formation.  But even though we don’t understand the non-rotating case very well, it is interesting to re-introduce rotation while maintaining horizontal homogeneity.  Adding rotation has a profound influence on the results — the model atmosphere fills up with tropical cyclones!  Some colleagues suggest referring to this system as TC World;  others suggest Diabatic Ekman Turbulence.  I’m going to stick with Rotating Radiative-Convective Equilibrium, or Rotating RCE for short.

You can include rotation while retaining horizontal homogeneity by adding a Coriolis force of fixed strength, independent of latitude.  In fact, one typically ignores the vertical component of the Coriolis force and simply adds terms to the horizontal equations of motion that, in isolation, would cause the horizontal winds to rotate at a fixed rate $f$, the Coriolis parameter.  (This geometry is referred to as the $f$-plane in textbooks and articles on geophysical fluid dynamics.).

Wenyu Zhou has been studying Rotating RCE in collaboration with several of us at GFDL. The first paper on this work is Zhou et al, 2014. The animation above is the near-surface wind speed from one of the simulations analyzed in this paper. Red corresponds roughly to hurricane strength winds.  $f = 2\Omega \sin(\theta)$ with $\theta = 20^\circ$ latitude and $\Omega$ is the magnitude of the angular velocity of the Earth.  Surface temperatures are fixed at 300K.  A month of simulation is shown, after several months of equilibration starting from an initial condition with no TCs present.

In studies of RCE, we often push the horizontal grid down to 1 or 2 km to help in explicitly simulating at least the largest convective plumes that extend to the tropopause.  In this paper we use a much coarser resolution, 25km, to the consternation of some reviewers  — we simply take a global atmospheric model with 25 km resolution and place it in this idealized $f$-plane geometry.  The model includes a a sub-grid closure scheme for moist convection.  The number of grid points in the horizontal is 800×800, producing a 20,000km square domain.  This is not meant as a model of a little patch of the atmosphere!  We are, of course, interested in how a model with 1 or 2 km grid would behave, but that would be computationally expensive for us even in a smaller domain barely large enough to contain a few storms — we want to have enough storms in the domain that we can study things like how the average spacing between storms varies with rotation rate or SST (this distance increases with decreasing f and with increasing SST.)  But we are also especially interested in how our global model, which simulates the geographical and seasonal distribution of TC genesis rather well (post #2), behaves in this idealized geometry.  I was involved in an earlier paper taking the same approach of placing a global model in this idealized $f$-plane geometry, Held and Zhao 2008, but with even coarser resolution.  That paper did not create much of a stir.  This approach gets more interesting (and the review process becomes a bit less painful) when the global model that we start with has TC statistics that look realistic. Increasing computer power should make exploration of this kind of rotating moist-convective turbulence more common.

In a recent paper Khairoutdinov and Emanuel 2013 have generated simulations with 3 km resolution that produce multiple storms that qualitatively resemble the result shown above.  They make the computation tractable by increasing $f$ by an order of magnitude compared to Earth-like values, resulting in storms small enough that you can get into this multiple storm regime much more easily.

You can get a sense from the video that the model does produce storms with a relatively well-defined radius of maximum winds.  Wenyu describes how this internal storm scale, despite our low resolution, changes systematically with model parameters.  This is obviously one place where there is likely to be important sensitivity to resolution.  But we also find that the size of the domain, if too small, can modify the sensitivity of this radius of maximum winds to other parameters by not allowing the storm to settle into its preferred horizontal structure.  A nice comparison of theories for mature TC structure with numerical simulations, including Rotating RCE, can be found in the recent thesis of Daniel Chavas 2013.

The most interesting qualitative result to me is simply that in this homogeneous system the natural equilibrated state is an atmosphere filled with TCs.  In reality, and in this model when run over realistic boundary conditions on a rotating sphere, TCs are very far from being so all-pervasive.  This seems partly to be due to the very long lifetimes of the vortices in this model.  Nearly all of the storms in the video survive over the month shown.  There is very little merging of vortices.  And there is, by construction, no movement of vortices over land, cutting off their energy supply, or poleward drift into midlatitudes followed by being torn apart by jets and extratropical storms. (This poleward drift is due in large part to the increase in strength of the Coriolis parameter with latitude on a rotating sphere, a gradient not present in our $f$-plane setup.)  The storms in rotating RCE just pile up, to a first approximation, until the occasional decay/merger is balanced by the occasional new storm managing to squeeze in and grab enough of the energy source at the surface.

In addition, if one can get far enough away from the influence of other storms the homogeneous environment here is always conducive to the genesis of new storms.  There are no strong vertical shears of the large scale horizontal winds, or large-scale dry-air intrusions, and no SSTs that are too cold to allow convection up to the tropopause.  All of these suppression mechanisms result from large-scale horizontal inhomogeneities.

Rotating RCE produces a distinctive kind of turbulence, dominated by vortices of one sign that are strongly dissipative and dependent for their survival on continuous access to their energy source.  Are there analogies to turbulent flows that arise in other contexts?

Whenever setting up an idealized model like this you have to ask if detailed study would really help us understand nature.  My intuition is that Rotating RCE will turn out to be very valuable — especially if we can devise clean ways of systematically reintroducing relevant inhomogeneities using the homogeneous case as a starting point.

[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]

### 5 Responses to “43. Rotating radiative-convective equilibrium”

1. Hank Roberts says:

Any similarity to this?, I stumbled over this today:
http://cimss.ssec.wisc.edu/goes/blog/
Low-level “barrier jet” along the southeast coast of Greenland
December 29th, 2013

” It is interesting to note that a secondary area of low pressure was seen rotating around the primary low, and appeared to be rapidly intensifying judging from the quick development of a “corkscrew” appearance on the water vapor imagery near the end of the animation….”

2. Frank says:

When you fix the surface temperature 300 degK, are you preventing the ocean at center of these storms from being cooled by evaporation and a clouds? (If I remember correctly, tropical cyclones leave a trail of depressed SSTs behind them.)

How important are tropical cyclones to surface energy balance in the tropics? From rainfall, one can calculate the amount of latent heat removed from the surface (probably in the vicinity of 50% more than the average of roughly 80 W/m2). What fraction of tropical rainfall comes from tropical cyclones?

It would be interesting to know if a horizontal temperature gradient in SST would eliminate the cyclones and produce bulk motion like the Hadley circulation.

• Isaac Held says:

Frank,
Your first question occurs to a lot of people when they see this. We’re looking into similar simulations in which surface temperatures are allowed to respond. We’ve done this in our aqua planet global model (see previous post) as well by looking at the simulations as a function of the depth of the heat capacity of the surface. We are still sorting out the results, which are not entirely intuitive.

See Jiang and Zipser, 2010 for one study of the fraction of rainfall that is due to tropical cyclones.

You can refer to the previous post to see how the same atmospheric model behaves with inhomogeneous SSTs on the sphere. We have some inhomogeneous simulations on the f-plane but we are not happy that we have the best configuration yet.