Linear waves in a one-layer ocean model with thermodynamics

The simplest member of a family of ocean models with inhomogeneous layers is used to study the physics of an active mixed layer overlying a quiescent denser layer. The dynamical variables of the system are the depth-averaged horizontal velocity u and buoyancy v fields and the layer thickness h. The potential vorticity q is not conserved in these models, unless v is constant. An integral of motion quadratic in the deviation from a steady and symmetric basic state is derived and shown not to be positive definite; that is, there is no formal stability theorem (except in the case of uniform flow). This is due to a term proportional to the square of the distance between the isolines of q and v, which is a measure of the nonconservation of q. A reference state without mean currents must have Theta H-2 = constant, where Theta and H are the buoyancy and layer thickness, respectively, in that state. Linear waves superimposed on this basic state are Poincare and Rossby waves, analogous to those of a homogeneous layer problem but with a different topography. In addition, there is a force-compensating mode (FCM), which corresponds to changes in buoyancy and layer thickness that represent a vanishing contribution to the depth-averaged pressure force. The phase speed of gravity waves (in the absence of rotation effects) is c = root Theta H. Rossby waves are driven by the gradient of f/c(2) (where f is the Coriolis parameter) rather than by the gradient of f/H, as they are in the homogeneous layer case. The quadratic integrals of motion are used to classify the types of instability. First, this is done in an a priori sense, looking for properties of an unstable basic flow that constrain the structure of growing perturbations. Second, this is done in an a posteriori sense, through the evaluation of the rate change of selected parts of those integrals of motion, as a function of the perturbation fields. It is stressed that for this method to give unambiguous results it is necessary that the integrals of motion be quadratic, to lowest order, in the perturbation. Unlike the case of the classical shallow water equations, a uniform flow in a channel may be unstable, if the buoyancy increases toward its right (left) in the northern (southern) hemisphere. In the f plane this necessary instability condition corresponds to mean flow in the direction opposite to that of the propagation of Rossby waves. A growing perturbation is very well represented by the combination of a Rossby wave and a FCM (from the system without currents) which interact due to the presence of the uniform flow.