Conditional Instability of the Second Kind

Conditional Instability of the Second Kind (CISK) is a landmark dynamical instability theory in tropical meteorology formulated by Jule G. Charney and Arnt Eliassen in 1964 (and independently by Katsuyuki Ooyama in 1964) to explain the genesis and intensification of synoptic-scale tropical depressions and hurricanes. In the late 1950s, linear stability analyses of a conditionally unstable tropical atmosphere—in which saturated air parcels are buoyant relative to their environment (∂θ_e*/∂z < 0)—revealed a fundamental scale-selection paradox: ordinary gravitational buoyancy instability ("conditional instability of the first kind") grows fastest at horizontal wavelengths of individual cumulonimbus clouds (1 to 10 km) with e-folding times of 15 to 30 minutes, rather than organizing into a balanced, warm-core cyclonic vortex spanning 200 to 1,000 km over several days. Charney and Eliassen resolved this dilemma by proposing that deep convective clouds and a balanced synoptic-scale cyclonic circulation engage in cooperative intensification: instead of competing for the same local parcel buoyancy, the balanced cyclone uses surface frictional Ekman pumping to converge water vapor over hundreds of kilometers and feed the cumulus ensemble, while the resulting tropospheric latent heat release lowers central surface pressure and spins up the balanced vortex.

Failure of Linear Conditional Instability of the First Kind

During the 1950s, meteorologists sought to model tropical cyclone formation by linearizing the primitive equations around a resting, conditionally unstable tropical atmosphere where saturated equivalent potential temperature decreases with height (∂θ_e*/∂z < 0, corresponding to a lapse rate Γ between the dry adiabatic lapse rate Γ_d = g/c_p = 9.8 K km⁻¹ and the moist pseudoadiabatic lapse rate Γ_m ≈ 4.5–6.0 K km⁻¹). In early models, the entire ascending branch of the perturbation was assumed to be 100% saturated so that heating was proportional to resolved vertical velocity w via a negative effective static stability S_eff < 0.

When solved as an eigenvalue problem for normal modes proportional to e^{σt + ikx}, the growth rate σ(k) increased monotonically with horizontal wavenumber k, reaching its maximum at small cumulus scales (λ = 2π/k ≈ 2–10 km) where hydrostatic and Coriolis effect rotational constraints are negligible. Consequently, numerical simulations using direct grid-scale latent heating collapsed into isolated thunderstorm towers rather than developing a synoptic-scale hurricane.

Charney and Eliassen's Cooperative Scale Separation

Charney and Eliassen (1964) recognized that the tropical troposphere is not uniformly saturated on the 500-km scale of a nascent tropical depression; instead, buoyant updrafts occupy only a small fractional area (σ_c ≈ 1%–5%) consisting of narrow cumulonimbus cores embedded within a broader, unsaturated synoptic environment. Because the ambient tropical atmosphere is stably stratified with respect to unsaturated vertical displacements (∂θ/∂z > 0, dry buoyancy frequency N² > 0), a balanced synoptic-scale vortex experiences positive static stability, whereas individual cumulonimbus clouds release latent heat on convective timescales of ~30 minutes.

Charney and Eliassen defined Conditional Instability of the Second Kind as a cooperative macro-scale instability coupling these two distinct scales:

1. Synoptic Scale → Cumulus Scale (Moisture Supply): A weak pre-existing low-level cyclonic vorticity anomaly (ζ_g > 0) experiences surface aerodynamic drag in the planetary boundary layer. Through frictional cross-isobar inflow (Ekman transport), moist boundary-layer air converges horizontally toward the vortex center and is pumped upward at the top of the boundary layer at the Charney–Eliassen Ekman pumping velocity w_E = (H_E / 2) ζ_g, supplying water vapor to sustain deep cumulus convection.
2. Cumulus Scale → Synoptic Scale (Vortex Stretching): The ensemble of deep cumulonimbus clouds condenses the converged water vapor and releases latent heat Q_c throughout the middle and upper troposphere. This diabatic heating warms the atmospheric column, induces mid-level inflow and vortex-tube stretching (f ∂w/∂z > 0), and intensifies low-level geostrophic vorticity ζ_g, which in turn drives stronger frictional Ekman moisture convergence.

Two-Level Balanced Quasi-Geostrophic Equations

In their 1964 paper in the Journal of the Atmospheric Sciences, Charney and Eliassen formulated CISK using a two-level balanced model on an f-plane (Coriolis parameter f = 2Ω sin φ ≈ 3.77 × 10⁻⁵ s⁻¹ at 15° N) in pressure coordinates, with upper-tropospheric level 1 (p₁ = 250 hPa), mid-tropospheric interface level 2 (p₂ = 500 hPa), lower-tropospheric level 3 (p₃ = 750 hPa), and the top of the frictional Ekman boundary layer at level 4 (p₄ ≈ 900–950 hPa).

Within the free troposphere above the boundary layer, the linearized vorticity equations at levels 1 and 3 and the hydrostatic thermodynamic energy equation at middle level 2 are:

∂ζ₁ / ∂t = f ( ω₂ / Δp ) , ∂ζ₃ / ∂t = f ( ω₄ − ω₂ ) / Δp

∂ / ∂t ( ψ₁ − ψ₃ ) + S₂ ω₂ = − ( R / f c_p ) Q₂

where ψ₁ and ψ₃ are the geostrophic streamfunctions (ζ_j = ∇²ψ_j), ω = dp/dt is the pressure vertical velocity (ω < 0 for upward motion), Δp = 500 hPa is the half-depth of the troposphere, S₂ = −(R T₂ / f p₂ θ₂) (∂θ / ∂p)₂ > 0 is the dry static stability parameter, and Q₂ is the cumulus diabatic heating rate per unit mass (J kg⁻¹ s⁻¹).

Boundary-Layer Ekman Pumping and Moisture-Convergence Closure

At the base of the free troposphere (level 4), upward vertical motion ω₄ is driven by frictional convergence in the planetary boundary layer according to the Charney–Eliassen (1949) bottom Ekman pumping relation:

ω₄ = − ρ₄ g w_E = − ρ₄ g ( H_E / 2 ) ζ₄ ≈ − E f Δp ∇²ψ₃

where H_E = (2A_z / f)^1/2 is the Ekman boundary-layer depth and E is the dimensionless Ekman number characterizing boundary-layer friction.

To close the system without resolving individual convective clouds, Charney and Eliassen introduced the CISK moisture-convergence parameterization: the vertically integrated cumulus latent heating rate is proportional to the total water vapor converged vertically through the top of the Ekman boundary layer and along the lower troposphere:

Q₂ = − η ( L_v q₄ ω₄ + L_v q₂ ω₂ ) / Δp ⇒ ( R / f c_p ) Q₂ = − μ S₂ ( ω₂ + ω₄ ) / 2

Here L_v = 2.50 × 10⁶ J kg⁻¹ is the latent heat of vaporization, q is specific humidity, and μ = (L_v Δq) / (c_p Δθ) is the dimensionless diabatic heating efficiency parameter representing the ratio of cumulus latent heat release to adiabatic cooling.

Growth Rate Relation and Synoptic Scale Selection

Substituting the CISK heating closure and the Ekman pumping boundary condition ω₄ into the two-level vorticity and thermodynamic equations yields the characteristic growth rate σ(k) for a disturbance of horizontal scale L = π / k relative to the internal Rossby radius of deformation L_D = (f⁻¹) (S₂ Δp / 2)^1/2:

σ(k) = ( E f / 2 ) · [ ( 2μ − 1 ) k² L_D² − 1 ] / [ 1 + ( 1 − μ ) k² L_D² ]

This analytic dispersion relation establishes two foundational results:

1. Threshold for Instability vs. Ekman Spin-Down: In a dry atmosphere (μ = 0), boundary-layer friction acts as a sink of balanced circulation, causing the vortex to decay at the Ekman spin-down rate σ ≈ −Ef / 2 (τ_spin ≈ 2–4 days). When the moisture parameter exceeds the threshold μ > 1/2 (specifically 1/2 < μ < 1 for balanced synoptic modes), boundary-layer friction switches from a damping agent to the driver of instability (σ > 0).
2. E-Folding Intensification Time Scale: Evaluating σ(k) at horizontal scales of 100 to 300 km for tropical moisture parameters (μ ≈ 0.7–0.9, f ≈ 3.8 × 10⁻⁵ s⁻¹) yields e-folding amplification times σ⁻¹ of 1 to 3 days, matching the observed intensification rate of nascent tropical depressions into hurricanes.

Extension to Wave-CISK (1970–1974)

In the early 1970s, Yoshikazu Hayashi and Richard S. Lindzen (1974) extended the CISK concept to the equatorial waveguide (where f = βy → 0), formulating Wave-CISK. In Wave-CISK, low-level moisture convergence is provided by the inviscid horizontal velocity convergence of vertically propagating internal gravity waves, Kelvin waves, and mixed Rossby–gravity waves. Although Wave-CISK offered an influential framework for convectively coupled equatorial waves and the Madden–Julian Oscillation (MJO), linear Wave-CISK models still favored the narrowest resolved horizontal scales unless vertical heating profiles were tuned.

Emanuel's WISHE Critique and Thermodynamic Equilibrium (1986–1994)

Starting in the mid-1980s, Kerry A. Emanuel (1986) and Richard Rotunno mounted an observational and physical critique of linear CISK as the primary mechanism for mature hurricane intensification:

1. Convective Quasi-Equilibrium in the Maritime Tropics: Dropsonde soundings across the tropical oceans (Xu & Emanuel, 1989) demonstrated that the tropical troposphere is maintained by deep convection close to moist convective quasi-equilibrium: the virtual temperature of a rising boundary-layer parcel closely tracks the ambient tropospheric virtual temperature profile. Consequently, converging existing boundary-layer air horizontally without increasing its equivalent potential temperature (θ_e) does not warm the tropospheric column relative to its environment.
2. Requirement for Surface Enthalpy Fluxes: To replace CISK for mature storms, Emanuel (1986) formulated the Wind-Induced Surface Heat Exchange (WISHE) mechanism, modeling a hurricane as a closed Carnot heat engine in which stronger surface winds |u₁₀| accelerate sea-to-air latent heat transfer (E_flux = ρ C_E |u₁₀| [q*(T_SST) − q_b]), raising boundary-layer θ_e inside the eyewall. Nonetheless, Charney and Eliassen's (1964) frictional Ekman convergence mechanism remains central to tropical meteorology: modern boundary-layer theory confirms that frictional Ekman inflow plays an essential dynamical role in spinning up tangential winds near the radius of maximum wind (Smith, Montgomery, & Nguyen, 2009) and organizing moisture inside nascent tropical disturbances.

What is the difference between Conditional Instability of the First Kind and the Second Kind?

Conditional Instability of the First Kind (CIFK) is ordinary parcel buoyancy instability in a conditionally unstable atmosphere (Γ_m < Γ < Γ_d), operating on the scale of individual cumulonimbus clouds (1–10 km) over 15–30 minutes. Conditional Instability of the Second Kind (CISK) is a cooperative macro-scale instability (100–1,000 km, 1–3 days) in which a balanced cyclonic vortex supplies low-level moisture via frictional Ekman pumping to a cumulus ensemble, and the resulting collective latent heat release drives vortex stretching that intensifies the balanced cyclone.

Why does surface friction promote growth in CISK when friction normally damps fluid motion?

In a dry barotropic vortex, surface friction drives cross-isobar Ekman inflow and upward Ekman pumping w_E > 0, forcing adiabatic cooling and spin-down over 2 to 4 days. In a moist tropical atmosphere (μ > 1/2), upward Ekman velocity w_E lifts water-vapor-rich air past its lifting condensation level, releasing more latent heat than is lost to adiabatic expansion and turning boundary-layer inflow into a net generator of cyclonic vorticity.

How does WISHE differ from CISK in explaining hurricane intensification?

CISK assumes that the tropical atmosphere contains a reservoir of convective instability that can be tapped by converging pre-existing boundary-layer water vapor horizontally via frictional Ekman inflow. WISHE (Wind-Induced Surface Heat Exchange, Emanuel 1986) recognizes that the tropical troposphere is nearly neutral to ambient boundary-layer parcels, so hurricane intensification requires raising the moist enthalpy (θ_e) of boundary-layer air through wind-speed-dependent evaporation from the warm ocean surface.

1. Charney, J. G., & Eliassen, A. (1964). "On the growth of the hurricane depression." Journal of the Atmospheric Sciences, 21(1), 68–75. https://doi.org/10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2021%3C0068:OTGOTH%3E2.0.CO;2)
2. Ooyama, K. (1969). "Numerical simulation of the life cycle of tropical cyclones." Journal of the Atmospheric Sciences, 26(1), 3–40. https://doi.org/10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2026%3C0003:NSOTLC%3E2.0.CO;2)
3. Lindzen, R. S. (1974). "Wave-CISK in the tropics." Journal of the Atmospheric Sciences, 31(1), 156–179. https://doi.org/10.1175/1520-0469(1974)031<0156:WCITT>2.0.CO;2031%3C0156:WCITT%3E2.0.CO;2)
4. Emanuel, K. A. (1986). "An air-sea interaction theory for tropical cyclones. Part I: Steady-state maintenance." Journal of the Atmospheric Sciences, 43(6), 585–605. https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2043%3C0585:AASITF%3E2.0.CO;2)
5. Smith, R. K., Montgomery, M. T., & Nguyen, S. V. (2009). "Tropical cyclone spin-up revisited." Quarterly Journal of the Royal Meteorological Society, 135(642), 1321–1335. https://doi.org/10.1002/qj.428