I have gone through both the references - The effect of jet streak curvature on kinematic fields and the background reference in that paper - Isolation of the inertial gravity component in a nonlinear atmospheric model and neither of those references mention any connection to a Cartesian based NS and defining curvature in that coordinate space. To me the curvature problem in meteorology is best studied(for localized air flow) in a
Natural Coordinate System , What are natural coordinates ?
Given that premise I would like to talk about the Frenet Serret formula. In this context the natural coordinate system has three orthogonal basis vectors($e_s,e_n,e_z)$ where $e_s$ is the direction of the horizontal wind vector i.e.
$$v_h = |v_h| * e_s $$ and $e_s$ is just the well known Streamline and $e_n$ is the normal to the streamline. We also need to introduce the radius of curvature which is used in the Frenet Serret formulae. It must be noted that the curvature talked about in this answer is the extrinsic curvature
Because of the orthogonality $e_s \times e_n = e_z$
I will first address AtmosphericPrisonEscape's question then proceed to answer OP's question.
The Frenet Serret formula does include the rate of change of basis vectors. From a layman's perspective can you explain what is rate of change of basis vectors ? Well meteorological problems can be defined in a Cartesian basis (x,y,z) that is "fixed" with respect to the fixed stars. So this system can be regarded as an inertial system for a earthbound observer. In this system there are no "fictitious" forces that arise due to the particular choice of coordinate system. Details about this "fixed" inertial frame are provided in the reference cited at the end of the answer in Chapter 1.
Since meteorological observations are carried out in a rotating coordinate system they are best described in a coordinate system that rotates with the earth. So Cartesian system moving point to point the basis vector remains fixed and not does not vary. OTOH in a earth like system (such as a sphere) we generally describe in terms of curvilinear coordinates whose basis vectors vary point to point.
Mathematically in a stationary Cartesian system $ \frac{de_x}{dx} = 0 $ because the basis vectors are independent of position.
In a general curvilinear system we let go of these assumptions and introduce the idea of curvature. So from a Frenet Serret perspective
$\frac{\partial e_s}{\partial n} = \frac{e_n}{R_n}$ and similar other terms.
Now to answering OP's question.
If we take this equation and differentiate it wrt time
$$v_h = |v_h| * e_s $$
we get
$$ \frac{dv_h}{dt} = e_s \frac{ d|v_h|}{dt} + |v_h|\frac{de_s}{dt}$$
As one can see here I am explicitly differentiating the basis vector $e_s$ wrt time.
So the first term on the RHS is known as the tangential acceleration(because it is in the direction of the streamline) and the second term after a usage of the Frenet Serret formula(and adjusting few terms) can be equated to
$$|v_h|\frac{de_s}{dt} = e_n \frac{v^2_h}{R_t}$$ and $R_t$ is the radius of Trajectory
The second term is known as the centripetal acceleration and is perpendicular to the streamline.
Now armed with this information we can attempt to answer all of OP's questions.
1) Curvature of jet streaks can be used to study areas of confluence and diffluence.
All one has to do is take the divergence of the horizontal wind vector. Keeping in mind that the gradient and the wind vector need to be expressed in natural coordinates. In the area of diffluence the radius of curvature is positive and negative in the area of confluence.
2) Finally the issue of Cyclostrophic balance can be easily derived by transforming the NS equation into natural coordinates as shown in this ESSE answer Eye of a tornado
I am not going to derive it but just give OP enough clues so OP can derive it.
The normal component of the NS equation is equated to the normal component of the pressure gradient. I have already given the equation for the normal component of the NS equation above. If we equate those two we get the cyclostrophic balance as shown below
$$|v_h|\frac{de_s}{dt} = e_n \frac{v^2_h}{R_t}$$
$$\frac{v^2_h}{R_t} = -\frac{1}{\rho}\frac{\partial p}{\partial n}$$
As one can see the curvature of the jet streak influences the acceleration by the centripetal force.
References - Dynamics of the atmosphere : A course in theoretical meteorology Chapter 8.