Introduction to PDEs and waves for the atmosphere and ocean.

*(English)*Zbl 1278.76004
Courant Lecture Notes in Mathematics 9. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 0-8218-2889-4/pbk). ix, 234 p. (2003).

The most important features that distinguish fluid flow in the atmosphere and ocean are the effects of rotation and stratification. These lecture notes focus on the rotating Boussinesq equations, the simplest equations that capture both features. The author presents rigorous mathematical theory concerning the rotating Boussinesq equations and offers deep insights into the observed phenomena of atmospheric and oceanic motions. The contribution of these notes to the modern literature on the dynamics of geophysical flows is very valuable and unique. In the author’s own words, “The lectures emphasize the serendipity between modern applied mathematics and geophysical flows in the style of modern applied mathematics where rigorous analysis [and] asymptotic, qualitative, and numerical modeling all interact.” This text is essentially self-contained and does not require prior knowledge of fluid dynamics, although a previous book by the author and A. L. Bertozzi [Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (2002; Zbl 0983.76001) would certainly help familiarize the reader with incompressible flows. These lecture notes should be of interest to anyone in the community of atmosphere/ocean science, from graduate students to advanced researchers.

The text is divided into 9 chapters. Chapter 1 introduces the rotating Boussinesq equations. §1.1 points outs an elementary exact solution with hydrostatic balance and writes the equations for the perturbation about a mean state in hydrostatic balance. §1.2 studies several exact two-dimensional solutions of the rotating Boussinesq equations. §1.3 builds elementary exact solutions that illustrate the important effect of gravity. §1.4 introduces the basic model solutions of jet flows with rotation and stratification. §1.5 solves the linearized primitive equations and represents their solutions as infinite combinations of eigenmodes of the shallow water equations.

Chapter 2 links some remarkable features of stratified flows with special solutions of the rotating Boussinesq equations. §2.1 derives the energy principle, namely the conservation of energy in the absence of forcing and dissipation. §2.2 provides the Boussinesq equations for the gradient of velocity and vorticity and proves Ertel’s theorem, an identity stating that the component of vorticity along the density gradient cannot be generated or destroyed. At the end, a family of special solutions is given. §2.3 through §2.6 examine nonlinear plane waves and other specific examples of the special solutions in §2.2.

Chapter 3 investigates the issue of whether the basic pendulum flows are unstable to perturbations and whether these perturbations can create local unstable overturnings in the fluid flow with strong initial stratification. §3.1 rewrites the Boussinesq equations in vorticity stream form and uses a novel mean Lagrangian coordinate to study perturbations. In the mean Lagrangian coordinate system both linear stability analysis and numerical computations become very elegant. The linear stability analysis reduces to studying the instability of an infinite number of decoupled Hill equations through Floquet theory. §3.2 presents numerical results on the nonlinear instability of pendulum flows and overturning. §3.3 investigates the instability properties of elementary linear shear flows. §3.4 recalls several basic facts in Floquet theory.

Chapter 4 is devoted to the rotating shallow water theory, which provides an appropriate approximation for atmospheric and oceanic motions in the midlatitudes, where the effects of the earth’s rotation are important. §4.1 introduces the rotating shallow water equations (RSWE). §4.2 and §4.3 derive the conservation laws for potential vorticity and energy. §4.4 studies solutions of the linearized RSWE. In particular the inertial-gravity waves and general plane wave solutions are investigated. §4.5, §4.6 and §4.7 nondimensionalize the RSWE and provide a formal derivation of the quasi-geostrophic (QG) equations from the RSWE as the Rossby number goes to zero. §4.8 rewrites the RSWE into a form that allows easy comparison with the QG equations, namely the model RSWE. §4.9, §4.10 and §4.11 are devoted to a rigorous proof for the convergence of the model RSWE to the QG equations. §4.9 derives a basic energy estimate and provides the mathematical tools needed in the proof. §4.10 states the rigorous convergence result as a theorem and outlines the major steps in the proof, and §4.11 actually presents the detailed proof.

Chapter 5 develops the theory of dispersive waves for linear, weakly nonlinear and nonlinear geophysical equations, the major tool being the WKB approximation. §5.1 studies the linear waves for the QG equation. §5.2 discusses the general properties of dispersive waves, first for a scalar wave equation and then for systems. §5.3 interprets the group velocity of dispersive waves. In §5.4, the long time behavior of solutions to linear wave equations is sought. The method of stationary phase is stated and applied to the scalar wave equation. In §5.5, the WKB approximation method is applied first to a linear first-order system with coefficients that vary slowly with space and time and then to the midlatitude quasi-geostrophic equations. The WKB procedure leads to simplified equations for the wave amplitudes and phases. The equation for the leading order amplitude, or the eikonal equation, is rewritten in terms of characteristics. In §5.6, the eikonal equation resulting from the scalar wave equation with rapidly oscillating phase is revisited and its solution is evaluated beyond the phase shock set using the stationary phase principle. §5.7 applies the WKB procedure to the first-order, constant coefficient system with a quadratic nonlinear term. The nonlinear term causes energy exchange among waves and creation of new waves not present in the initial data. §5.8 is devoted to the nonlinear WKB theory for the Boussinesq equations.

Chapter 6 is concerned with the simplified equations that describe fluid motion with strong stratification and weak rotation. §6.1 provides formal derivations of the layered 2D Euler equations from the nondimensional Boussinesq equations and the layered 2D Navier-Stokes equations from the nondimensional viscous Boussinesq equations with rotation in the limit of small Froude number. §6.2 derives the vorticity-stream form of these equations. §6.3 first lists the major features of the experimental observations in decaying strongly stratified turbulence and then demonstrates that the exact solutions of the simplified dynamics capture the listed features of experimental observations.

Chapter 7 studies geophysical flows subject to the combined effects of rotation and density stratification. The author shows that when the Rossby and Froude numbers are both small and comparable in magnitude, the rapidly rotating Boussinesq equations (RBE) for stably stratified flows converge to the stratified quasi-geostrophic equations (QGE). This chapter is divided into seven sections. §7.1 is an introduction. §7.2 extends Ertel’s theorem to the absolute vorticity, which takes into account the effect due to the rotating reference frame. It is also shown here that the pressure in the RBE can be recovered from the velocity and density gradients by solving a Poisson equation. §7.3 nondimensionalizes the RBE. §7.4 is devoted to a formal derivation of the QGE from the RBE in the asymptotic limit of small Rossby and Froude numbers. §7.5 states as a theorem the rigorous result on the convergence of the RBE to the QGE and outlines the major steps in the proof. §7.6 derives a basic energy estimate for the RBE and an elliptic regularity estimate for the Poisson equation. §7.7 provides the rigorous proof of the theorem. The proof follows the same strategy as in Chapter 4. The new issue is the need to derive estimates for the pressure. An elliptic regularity estimate for the Poisson equation is derived for this purpose.

Chapter 8 introduces the theory of averaging over fast waves for geophysical flows. §8.1 nicely summarizes the major content of this chapter. Many problems in geophysical flows involve the interaction of different fluid-dynamic mechanisms that evolve over two different time scales. In this chapter, the author discusses these phenomena for the rapidly rotating shallow water equations (RSWE) and the rapidly rotating Boussinesq equations. In §8.2, the author motivates these issues for RSWE. In §8.3, the author develops an abstract, formal two-time averaging principle, which is then applied to a general family of ODE to illustrate the general structure of the limit. §8.4 presents applications of this principle for the ODE generated by the exact local solution procedure for the Boussinesq equations developed in Chapter 2. In §8.5, the general theory of fast-wave averaging is applied to the quasi-geostrophic limit of RSWE with the motivation of understanding nonlinear Rossby adjustment in a fundamental mathematical fashion. In the last section of this chapter, the ideas for unbalanced initial data are applied to both the low Froude number and low Froude, low Rossby number limits of the Boussinesq equations.

Chapter 9 focuses on wave-like solutions of several special rotating Boussinesq equations modeling the equatorial atmosphere and ocean. For equatorial geophysical flows, the vanishing of the tangential projection of the Coriolis force is the main new feature. §9.1 studies three types of waves to the equatorial shallow water (ESW) equations. §9.2 is concerned with wave solutions of the linearized hydrostatic primitive equations in the troposphere with source terms. These equations are reduced to a countable number of constant-coefficient dispersive systems through the use of parabolic cylinder functions. This section ends with plots of the dispersive relations and structures of equatorial waves. §9.3 is on the nonlinear equatorial long-wave equations (NLELWE). §9.4 introduces a simple model for the steady circulation of the equatorial atmosphere, namely a linearized primitive equation with heating and radiative damping. Different types of waves are analyzed for symmetric and anti-symmetric heating.

The text is divided into 9 chapters. Chapter 1 introduces the rotating Boussinesq equations. §1.1 points outs an elementary exact solution with hydrostatic balance and writes the equations for the perturbation about a mean state in hydrostatic balance. §1.2 studies several exact two-dimensional solutions of the rotating Boussinesq equations. §1.3 builds elementary exact solutions that illustrate the important effect of gravity. §1.4 introduces the basic model solutions of jet flows with rotation and stratification. §1.5 solves the linearized primitive equations and represents their solutions as infinite combinations of eigenmodes of the shallow water equations.

Chapter 2 links some remarkable features of stratified flows with special solutions of the rotating Boussinesq equations. §2.1 derives the energy principle, namely the conservation of energy in the absence of forcing and dissipation. §2.2 provides the Boussinesq equations for the gradient of velocity and vorticity and proves Ertel’s theorem, an identity stating that the component of vorticity along the density gradient cannot be generated or destroyed. At the end, a family of special solutions is given. §2.3 through §2.6 examine nonlinear plane waves and other specific examples of the special solutions in §2.2.

Chapter 3 investigates the issue of whether the basic pendulum flows are unstable to perturbations and whether these perturbations can create local unstable overturnings in the fluid flow with strong initial stratification. §3.1 rewrites the Boussinesq equations in vorticity stream form and uses a novel mean Lagrangian coordinate to study perturbations. In the mean Lagrangian coordinate system both linear stability analysis and numerical computations become very elegant. The linear stability analysis reduces to studying the instability of an infinite number of decoupled Hill equations through Floquet theory. §3.2 presents numerical results on the nonlinear instability of pendulum flows and overturning. §3.3 investigates the instability properties of elementary linear shear flows. §3.4 recalls several basic facts in Floquet theory.

Chapter 4 is devoted to the rotating shallow water theory, which provides an appropriate approximation for atmospheric and oceanic motions in the midlatitudes, where the effects of the earth’s rotation are important. §4.1 introduces the rotating shallow water equations (RSWE). §4.2 and §4.3 derive the conservation laws for potential vorticity and energy. §4.4 studies solutions of the linearized RSWE. In particular the inertial-gravity waves and general plane wave solutions are investigated. §4.5, §4.6 and §4.7 nondimensionalize the RSWE and provide a formal derivation of the quasi-geostrophic (QG) equations from the RSWE as the Rossby number goes to zero. §4.8 rewrites the RSWE into a form that allows easy comparison with the QG equations, namely the model RSWE. §4.9, §4.10 and §4.11 are devoted to a rigorous proof for the convergence of the model RSWE to the QG equations. §4.9 derives a basic energy estimate and provides the mathematical tools needed in the proof. §4.10 states the rigorous convergence result as a theorem and outlines the major steps in the proof, and §4.11 actually presents the detailed proof.

Chapter 5 develops the theory of dispersive waves for linear, weakly nonlinear and nonlinear geophysical equations, the major tool being the WKB approximation. §5.1 studies the linear waves for the QG equation. §5.2 discusses the general properties of dispersive waves, first for a scalar wave equation and then for systems. §5.3 interprets the group velocity of dispersive waves. In §5.4, the long time behavior of solutions to linear wave equations is sought. The method of stationary phase is stated and applied to the scalar wave equation. In §5.5, the WKB approximation method is applied first to a linear first-order system with coefficients that vary slowly with space and time and then to the midlatitude quasi-geostrophic equations. The WKB procedure leads to simplified equations for the wave amplitudes and phases. The equation for the leading order amplitude, or the eikonal equation, is rewritten in terms of characteristics. In §5.6, the eikonal equation resulting from the scalar wave equation with rapidly oscillating phase is revisited and its solution is evaluated beyond the phase shock set using the stationary phase principle. §5.7 applies the WKB procedure to the first-order, constant coefficient system with a quadratic nonlinear term. The nonlinear term causes energy exchange among waves and creation of new waves not present in the initial data. §5.8 is devoted to the nonlinear WKB theory for the Boussinesq equations.

Chapter 6 is concerned with the simplified equations that describe fluid motion with strong stratification and weak rotation. §6.1 provides formal derivations of the layered 2D Euler equations from the nondimensional Boussinesq equations and the layered 2D Navier-Stokes equations from the nondimensional viscous Boussinesq equations with rotation in the limit of small Froude number. §6.2 derives the vorticity-stream form of these equations. §6.3 first lists the major features of the experimental observations in decaying strongly stratified turbulence and then demonstrates that the exact solutions of the simplified dynamics capture the listed features of experimental observations.

Chapter 7 studies geophysical flows subject to the combined effects of rotation and density stratification. The author shows that when the Rossby and Froude numbers are both small and comparable in magnitude, the rapidly rotating Boussinesq equations (RBE) for stably stratified flows converge to the stratified quasi-geostrophic equations (QGE). This chapter is divided into seven sections. §7.1 is an introduction. §7.2 extends Ertel’s theorem to the absolute vorticity, which takes into account the effect due to the rotating reference frame. It is also shown here that the pressure in the RBE can be recovered from the velocity and density gradients by solving a Poisson equation. §7.3 nondimensionalizes the RBE. §7.4 is devoted to a formal derivation of the QGE from the RBE in the asymptotic limit of small Rossby and Froude numbers. §7.5 states as a theorem the rigorous result on the convergence of the RBE to the QGE and outlines the major steps in the proof. §7.6 derives a basic energy estimate for the RBE and an elliptic regularity estimate for the Poisson equation. §7.7 provides the rigorous proof of the theorem. The proof follows the same strategy as in Chapter 4. The new issue is the need to derive estimates for the pressure. An elliptic regularity estimate for the Poisson equation is derived for this purpose.

Chapter 8 introduces the theory of averaging over fast waves for geophysical flows. §8.1 nicely summarizes the major content of this chapter. Many problems in geophysical flows involve the interaction of different fluid-dynamic mechanisms that evolve over two different time scales. In this chapter, the author discusses these phenomena for the rapidly rotating shallow water equations (RSWE) and the rapidly rotating Boussinesq equations. In §8.2, the author motivates these issues for RSWE. In §8.3, the author develops an abstract, formal two-time averaging principle, which is then applied to a general family of ODE to illustrate the general structure of the limit. §8.4 presents applications of this principle for the ODE generated by the exact local solution procedure for the Boussinesq equations developed in Chapter 2. In §8.5, the general theory of fast-wave averaging is applied to the quasi-geostrophic limit of RSWE with the motivation of understanding nonlinear Rossby adjustment in a fundamental mathematical fashion. In the last section of this chapter, the ideas for unbalanced initial data are applied to both the low Froude number and low Froude, low Rossby number limits of the Boussinesq equations.

Chapter 9 focuses on wave-like solutions of several special rotating Boussinesq equations modeling the equatorial atmosphere and ocean. For equatorial geophysical flows, the vanishing of the tangential projection of the Coriolis force is the main new feature. §9.1 studies three types of waves to the equatorial shallow water (ESW) equations. §9.2 is concerned with wave solutions of the linearized hydrostatic primitive equations in the troposphere with source terms. These equations are reduced to a countable number of constant-coefficient dispersive systems through the use of parabolic cylinder functions. This section ends with plots of the dispersive relations and structures of equatorial waves. §9.3 is on the nonlinear equatorial long-wave equations (NLELWE). §9.4 introduces a simple model for the steady circulation of the equatorial atmosphere, namely a linearized primitive equation with heating and radiative damping. Different types of waves are analyzed for symmetric and anti-symmetric heating.

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

76U05 | General theory of rotating fluids |

35Q35 | PDEs in connection with fluid mechanics |

76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76D33 | Waves for incompressible viscous fluids |

86A05 | Hydrology, hydrography, oceanography |

86A10 | Meteorology and atmospheric physics |