Fluid Mechanics

13.15. Baroclinic Instability

Weather maps at mid-latitudes invariably show the presence of wavelike horizontal excursions of temperature and pressure contours, superposed on eastward mean flows such as the jet stream. Similar undulations are also found in the ocean on eastward currents such as the Gulf Stream in the north Atlantic. A typical wavelength of these disturbances is observed to be of the order of the internal Rossby radius, that is, about 4000 km in the atmosphere and 100 km in the ocean. They seem to be propagating as Rossby waves, but their erratic and unexpected appearance suggests that they are not forced by any external agency, but are due to an inherent instability of mid-latitude eastward flows. In other words, the eastward flows have a spontaneous tendency to develop wavelike disturbances. In this section we shall investigate the instability mechanism that is responsible for the spontaneous meandering of large-scale eastward flows.

The poleward decrease of solar irradiation results in a poleward decrease of air temperature and a consequent increase of air density. An idealized distribution of the atmospheric density in the northern hemisphere is shown in Figure 13.31. The density increases northward due to the lower temperatures near the poles and decreases upward because of static stability. According to the thermal wind relation (13.15), an eastward flow (such as the jet stream in the atmosphere or the Gulf Stream in the Atlantic) in equilibrium with such a density structure must have a velocity that increases with height. A system with inclined density surfaces, such as the one in Figure 13.31, has more potential energy than a system with horizontal density surfaces, just as a system with an inclined free surface has more potential energy than a system with a horizontal free surface. Thus, this arrangement of atmospheric mass is possibly unstable because it can release the stored potential energy by means of an instability that would cause the density surfaces to flatten out. In the process, vertical shear of the mean flow U(z) would decrease, and perturbations would gain kinetic energy.

Instability of baroclinic flows that releases potential energy by flattening out constant density surfaces is called the baroclinic instability. The analysis provided here shows that the preferred scale of such unstable waves is indeed of the order of the Rossby radius, as observed for the mid-latitude weather disturbances. The theory of baroclinic instability was developed in the 1940s by Vilhem Bjerknes and others, and is considered one of the major triumphs of geophysical fluid mechanics. The presentation provided here is based on the review article by Pedlosky (1971).

Consider a simple basic state in which the density increases northward at a constant rate ∂ρ¯/∂y

and is stably stratified in the vertical with a uniform buoyancy frequency N. According to the thermal wind relation, the constancy of ∂ρ¯/∂y

requires that the vertical shear of the basic eastward flow U(z) also be constant. The β-effect is neglected here since it is not essential for the instability. (The β-effect does modify the instability, however.) This is borne out by the spontaneous appearance of undulations in laboratory experiments in a rotating annulus, in which the inner wall is maintained at a higher temperature than the outer wall. The β-effect is absent in such an experiment.

Perturbation Vorticity Equation

The equations for the total flow are the continuity equation (4.10), the horizontal momentum equations of (13.9) simplified for frictionless flow with negligible vertical velocity, vertical hydrostatic equilibrium (1.14), and the density equation (4.9). The total flow is assumed to be composed of an eastward wind U(z) in geostrophic equilibrium with the basic density structure ρ¯(y,z)

shown in Figure 13.31, plus perturbations:

u=U(z)+u′(x,t),v=v′(x,t),w=w′(x,t),ρ=ρ¯(y,z)+ρ′(x,t),andp=p¯(y,z)+p′(x,t).

(13.125)

The basic flow is in geostrophic and hydrostatic balance:

fU=−1ρ0∂p¯∂y,and0=−∂p¯∂z−ρ¯g.

(13.126, 13.127)

Eliminating the pressure, we obtain the thermal wind relation:

dUdz=gρ0f∂ρ¯∂y,

(13.128)

which requires the eastward flow to increase with height because ∂ρ¯/∂y>0

. Here, for simplicity, assume that ∂ρ¯/∂y

is constant, and that U = 0 at the surface z = 0. Thus, the background flow is:

U(z)=U0z/H,

where U₀ is the velocity at the top of the layer of interest, z = H.

Next form the vorticity equation by cross-differentiating and adding the frictionless horizontal momentum equations to eliminate the pressure. Then, use (4.10) to replace ∂u/∂x + ∂v/∂y with –∂w/∂z. The result is:

∂ζ∂t+u∂ζ∂x+v∂ζ∂y−(ζ+f)∂w∂z=0.

(13.129)

The development follows that leading to (13.92), except the β-effect is excluded here. Substitute the decompositions (13.125) into (13.129), drop nonlinear terms, and note that ζ = ζ′ because the basic flow U = U₀z/H has no vertical component of vorticity. After these steps, (13.129) becomes:

∂ζ′∂t+U∂ζ′∂x+−f∂w′∂z=0,

(13.130)

This is the perturbation vorticity equation, and it can be written in terms of p′.

Assume that the perturbations are large-scale and slow, so that the velocity is nearly geostrophic:

u′≅−1ρ0f∂p′∂y,andv′≅1ρ0f∂p′∂x.

(13.131)

from which the perturbation vorticity is found as:

ζ′=1ρ0f(∂2∂x2+∂2∂y2)p′=1ρ0f∇H2p′.

(13.132)

Next, develop an expression for w′ in terms of p′ using the density equation (4.9):

∂∂t(ρ¯+ρ′)+(U+u′)∂∂x(ρ¯+ρ′)+v′∂∂y(ρ¯+ρ′)+w′∂∂z(ρ¯+ρ′)=0.

Evaluate derivatives and linearize, to obtain:

∂ρ′∂t+U∂ρ′∂x+v′∂ρ¯∂y−ρ0N2w′g=0,

(13.133)

where N² is given by (13.1). The pressure is presumed to be hydrostatic, so the perturbation density ρ′ can be written in terms of p′ by using (1.14) and subtracting the background state (13.127) to reach:

0=−∂p′/∂z−ρ′g.

(13.134)

Substituting this into (13.133) leads to:

w′=−1ρ0N2[(∂∂t+U∂∂x)∂p′∂z−dUdz∂p′∂x],

(13.135)

where (13.128) has been used to write ∂ρ¯/∂y

in terms of the thermal wind dU/dz. Using (13.132) and (13.135), the perturbation vorticity equation (13.130) becomes:

(∂∂t+U∂∂x)[∇H2p′+f2N2∂2p′∂z2]=0.

(13.136)

This is the equation that governs quasi-geostrophic perturbations on an eastward flow U(z).

Wave Solution

Assume that (13.136) has traveling wave solutions,

p′=pˆ(z)exp{i(kx+ly−ωt)},

(13.137)

confined between horizontal planes at z = 0 and z = H that are unbounded in x and y. Real flows are likely to be bounded in the y direction, especially in a laboratory situation of flow in an annular channel, where the walls set boundary conditions parallel to the flow. Boundedness in y, however, simply sets up normal modes in the form sin(nπy/L), where L is the width of the channel. Each of these modes can be replaced by a periodicity in y.

Inserting (13.137) into (13.136), reduces (13.136) to an ordinary differential equation for pˆ

d2pˆdz2+α2pˆ=0,whereα2≡N2f2(k2+l2).

(13.138, 13.139)

The solution of (13.138) can be written as:

pˆ=Acosh[α(z−H/2)]+Bsinh[α(z−H/2)],

(13.140)

and is completely specified when the boundary conditions w′ = 0 at z = 0 & H are satisfied. The boundary conditions on p′ corresponding to those on w′ are found from (13.135) and U(z) = U₀z/H:

(∂∂t+U0zH∂∂x)∂p′∂z−U0H∂p′∂x=0atz=0andz=H.

In particular, these two boundary conditions are:

∂2p′∂t∂z−U0H∂p′∂x=0atz=0,and∂2p′∂t∂z+Uo∂2p′∂x∂z−U0H∂p′∂x=0atz=H.

Instability Criterion

Using (13.137) and (13.140), the two boundary conditions require:

A[αcsinhαH2−U0HcoshαH2]+B[−αccoshαH2+U0HsinhαH2]=0,andA[α(U0−c)sinhαH2−U0HcoshαH2]+B[α(U0−c)coshαH2−U0HsinhαH2]=0,

where c = ω/k is the eastward phase velocity.

This is a pair of homogeneous equations for the constants A and B. For non-trivial solutions to exist, the determinant of the coefficients must vanish. This gives, after some algebra, the phase velocity:

c=U02±U0αH(αH2−tanhαH2)(αH2−cothαH2).

(13.141)

Whether the solution grows with time depends on the sign of the radicand. The behavior of the functions under the radical sign is sketched in Figure 13.32. It is apparent that the first factor in the radicand is positive because αH/2 > tanh(αH/2) for all values of αH. However, the second factor is negative for small values of αH for which αH/2 < coth(αH/2). In this range the roots of c are complex conjugates, with c = U₀/2 ± ic_i. Because we have assumed that the perturbations are of the form exp(−ikct), the existence of a non-zero c_i implies the possibility of a perturbation that grows as exp(kc_it), and the solution is unstable. The marginal stability is given by the critical value of α satisfying:

αcH/2=coth(αcH/2),

whose solution is α_cH = 2.4, so the flow is unstable if αH < 2.4. Using the definition of α in (13.139), it follows that the flow is unstable if:

HN/f<2.4/k2+l2.

Since all values of k and l are allowed, a value of k² + l² low enough to satisfy this inequality can always be found. The flow is therefore always unstable (to low wave number disturbances). For a north-south wave number l = 0, instability is ensured if the east-west wave number k is small enough such that:

HN/f<2.4/k.

(13.142)

In a continuously stratified ocean, the speed of a long internal wave for the n = 1 baroclinic mode is c = NH/π, so that the corresponding internal Rossby radius is c/f = NH/πf. It is usual to omit the factor π and define the Rossby radius Λ in a continuously stratified fluid as:

Λ≡HN/f.

The condition (13.142) for baroclinic instability is therefore that the east-west wavelength be large enough so that λ > 2.6Λ.

However, the wavelength λ = 2.6Λ does not grow at the fastest rate. It can be shown from (13.141) that the wavelength with the largest growth rate is:

λmax=3.9Λ.

This is therefore the wavelength that is observed when the instability develops. Typical values for f, N, and H suggest that λ_max∼4000 km in the atmosphere and 200 km in the ocean, which agree with observations. Waves much smaller than the Rossby radius do not grow, and the ones much larger than the Rossby radius grow very slowly.

Energetics

The foregoing analysis suggests that the existence of planet-encircling weather waves is due to the fact that small perturbations can grow spontaneously when superposed on an eastward current maintained by the sloping density surfaces (Figure 13.31). Although the basic current does have a vertical shear, the perturbations do not grow by extracting energy from the vertical shear field. Instead, they extract their energy from the potential energy stored in the system of sloping density surfaces. The energetics of the baroclinic instability are therefore quite different than those of the Kelvin-Helmholtz instability where the perturbation Reynolds stress u′w′¯

extracts energy from the mean-flow's vertical shear. The baroclinic instability is not a shear-flow instability; the Reynolds stresses are too small because of the small w′ in quasi-geostrophic large-scale flows.

The energetics of the baroclinic instability can be understood by examining the equation for the perturbation kinetic energy. Such an equation can be derived by multiplying the equations for ∂u′/∂t and ∂v′/∂t by u′ and v′, respectively, adding the two, and integrating over the volume of the flow. Because of the assumed periodicity in x and y, the extent of the volume integration is appropriately confined to one wavelength in either direction. To complete this integration, the boundary conditions of zero normal flow on the upper and lower surfaces and periodicity in x and y are used repeatedly. The procedure is similar to that for the derivation of (11.88) and is not repeated here. The result is:

ddt(ρ02∫(u′2+v′2)dxdydz)=dKEdt=−g∫w′ρ′dxdydz,

where KE is the global perturbation kinetic energy. In unstable flows, dKE/dt must be greater than zero, which requires the volume integral of w′ρ′ to be negative. Denote the volume average of w′ρ′ by w′ρ′¯

. A negative w′ρ′¯

means that on average the lighter fluid rises and the heavier fluid sinks. By such an interchange the center of gravity of the system, and therefore its potential energy, is lowered. The interesting point is that this cannot happen in a stably stratified system with horizontal density surfaces; in that case an exchange of fluid particles raises the potential energy. Moreover, a basic state with inclined density surfaces (Figure 13.31) cannot have w′ρ′¯

< 0 if the particle excursions are only vertical. If, however, the particle excursions include northward and southward displacements, and fall within the wedge formed by the constant density lines and the horizontal (Figure 13.33), then an exchange of fluid particles takes lighter particles upward (and northward) and denser particles downward (and southward). Such an interchange would tend to make the density surfaces more horizontal, releasing potential energy from the mean density field with a consequent growth of the perturbation energy. This type of convection is called sloping convection. According to Figure 13.33 the exchange of fluid particles within this wedge of instability results in a net poleward transport of heat from the tropics, which serves to redistribute the larger solar heat received by the tropics.

In summary, baroclinic instability draws energy from the potential energy of the mean density field. The resulting eddy motion has particle trajectories that are oriented at a small angle with the horizontal, so that the resulting heat transfer has a poleward component. The preferred scale of the disturbance is the Rossby radius.