Working Through Stull's Boundary Layer Problems
Stull's textbook is the standard reference for boundary layer meteorology courses. The problem sets at the end of each chapter aren't trivial, and if you're grinding through them for a class, having a solid approach to the solutions will save you hours. I've worked through more of these than I care to count across several semesters. The solutions themselves are straightforward once you know what the problems are actually testing. Stull tends to build concepts incrementally — Chapter 2 convective boundary layer derivations lead directly into Chapter 3 flux calculations, and Chapter 4 similarity theory gets used constantly after that. The biggest mistake students make is treating each problem in isolation instead of recognizing the recurring frameworks. I spent way too long in grad school trying to re-derive Monin-Obukhov similarity from first principles for every stability case instead of just applying the universal functions. It works, but it's inefficient. Know when to derive and when to plug in.
Here's how I break down a typical Stull problem set: first, identify the regime. Is the PBL convective, neutral, or stable? That decision alone determines which equations are relevant. A neutral surface layer problem uses completely different assumptions than a CBL mixing problem, and mixing them up is the most common error I see. Second, list your knowns and unknowns with units. Third, check whether the problem asks for an estimate or a precise calculation — Stull distinguishes between the two, and the difference matters for grading. One specific thing that tripped me up repeatedly: the Obukhov length calculation. The formula itself is simple, L = -u*^3 / (k * gamma_w * theta_v), but the sign conventions and the virtual temperature correction get glossed over in the text. I remember working a problem where my L came out negative for a clearly stable night case because I'd dropped the virtual temperature adjustment and used raw potential temperature instead. The solver manual handles this correctly, but if you're working through it solo, double-check whether theta_v appears in your denominator or if theta is acceptable for the given problem conditions. For the convective boundary layer chapters, pay attention to the difference between the mixed-layer approximation and the full entrainment zone treatment. Stull presents both, and exams frequently ask you to choose between them. The mixed-layer approach assumes perfect mixing within the PBL and gives you bulk relationships quickly. The entrainment formulation is more accurate but requires knowing the entrainment rate w_e, which Stull gives as roughly 0.6 to 0.8 times the surface convective velocity scale w_star. Using w_e = 0 when the problem asks for entrainment effects is a sure way to lose points.
Stability classification is another area where beginners struggle. The Pasquill-Gifford categories map to specific ranges of L and u*, but Stull also uses his own dimensionless groups like z_i/L and z*/L. Getting comfortable converting between these systems early saves a lot of back-and-forth. I usually keep a one-page conversion sheet for this during problem sessions. When it comes to actually finding solutions, there are a few routes. The official instructor's solutions manual is the most reliable source. It walks through derivations step by step and shows the numerical work, which is where most mistakes happen. Student-shared PDFs exist online but they're hit or miss — I've seen at least three different versions with incorrect answers for Chapter 9 surface energy balance problems. If you're using a shared solution, verify any unexpected numbers against the textbook's worked examples in the preceding section.
Get the Full Details
Practical Tips for Working the Problems
Dimensional analysis should be your first step on every derivation problem. Stull's problems often ask for relationships in terms of specific variables, and checking dimensions before you finish catches algebra errors quickly. This habit cut my revision time on derivation problems from maybe 30 minutes to about 5 or 10. For numerical problems involving the surface layer, always verify that your stability parameter z/L falls within the validity range of the universal function you're using. The Businger-Dyer relations break down outside |z/L| < 1 for unstable and |z/L|
0.1 for stable conditions. I once ran a stable night case where z/L came out to about -0.8 and I used the unstable universal functions anyway because I didn't check. The wind profile error was noticeable but subtle enough that I missed it on the first pass. The turbulent kinetic energy budget problems in the later chapters are worth more time than their point value suggests. They tie together almost everything else in the book. If you understand how TKE is produced, dissipated, and transported in both convective and stable conditions, the rest of the problem set becomes significantly easier.
One limitation of relying heavily on Stull's solutions for study purposes: the book was published in 1988, and while the core physics hasn't changed, some of the parameterizations have been refined since then. The M-O similarity functions, for example, have undergone revisions from later authors like Wyngaard and Horst. If your course expects updates beyond the original text, supplement Stull with a more recent paper or textbook chapter on the specific topic. Don't assume Stull's numbers are the final word on anything post-1990. The entrainment parameter values Stull uses are also somewhat debated. Later work by Deardorff and others refined the w_e/w_* ratio, and some of his example calculations can look off if you compare them to modern LES results. This doesn't invalidate the textbook for course work, but it's worth noting if you ever need to defend your numbers in a research context.
