  We study the one-dimensional spin-1/2 Heisenberg model with antiferromagnetic
nearest-neighbor J_1 and next-nearest-neighbor J_2 exchange couplings in
magnetic field h. With varying dimensionless parameters J_2/J_1 and h/J_1, the
ground state of the model exhibits several phases including three gapped phases
(dimer, 1/3-magnetization plateau, and fully polarized phases) and four types
of gapless Tomonaga-Luttinger liquid (TLL) phases which we dub TLL1, TLL2,
spin-density-wave (SDW_2), and vector chiral phases. From extensive numerical
calculations using the density-matrix renormalization-group method, we
investigate various (multiple-)spin correlation functions in detail, and
determine dominant and subleading correlations in each phase. For the
one-component TLLs, i.e., the TLL1, SDW_2, and vector chiral phases, we fit the
numerically obtained correlation functions to those calculated from effective
low-energy theories of TLLs, and find good agreement between them. The
low-energy theory for each critical TLL phase is thus identified, together with
TLL parameters which control the exponents of power-law decaying correlation
functions. For the TLL2 phase, we develop an effective low-energy theory of
two-component TLL consisting of two free bosons (central charge c=1+1), which
explains numerical results of entanglement entropy and Friedel oscillations of
local magnetization. Implications of our results to possible magnetic phase
transitions in real quasi-one-dimensional compounds are also discussed.
